AmericanQuant

There's some wisdom here.  If just one person would make or break a school for you, you shouldn't go.  But if you are on the fence, knowing about the intentions of one person can make all the difference.

No, it's not fine to ask. There should always be someone else who could be your advisor.

You are definitely not disqualified from getting admitted to the best political science departments.  They are willing to take a chance on good people, and are understanding that people have made mistakes.  Unlike fields like physics, math, econ, etc., there aren't enough perfect applicants to go around, so compromises naturally arise.
 
If you have very good GRE scores and take tough courses, you should be in good shape.  For the GRE's, the better your quant score is the better.  For most applicants, they're just looking for some minimum level of quant competence (probably about 160 ish), but since you are going to not the best university and your grades are mixed, really concrete signals of competence are important.  Take lots of practice tests, and get study books if necessary.
 
The good grades in econ courses and quant polisci courses are positive signs.  Have you taken any math?  American political behavior and elections are very quant-heavy areas, so knowing calculus (and even some linear algebra) will be valuable. Do you have any interest in political methodology (i.e. applied statistics for political science problems)? If so, a good grade in real analysis would help your case, AND be valuable for your career.
 
Lastly, have you thought about taking some time off between college and grad school and working? Having your senior year grades available will probably help your case and a lot of people find it rewarding for other reasons. If you got a very good job in politics, it might also make you a better grad student.

You are definitely not disqualified from getting admitted to the best political science departments.  They are willing to take a chance on good people, and are understanding that people have made mistakes.  Unlike fields like physics, math, econ, etc., there aren't enough perfect applicants to go around, so compromises naturally arise.
 
If you have very good GRE scores and take tough courses, you should be in good shape.  For the GRE's, the better your quant score is the better.  For most applicants, they're just looking for some minimum level of quant competence (probably about 160 ish), but since you are going to not the best university and your grades are mixed, really concrete signals of competence are important.  Take lots of practice tests, and get study books if necessary.
 
The good grades in econ courses and quant polisci courses are positive signs.  Have you taken any math?  American political behavior and elections are very quant-heavy areas, so knowing calculus (and even some linear algebra) will be valuable. Do you have any interest in political methodology (i.e. applied statistics for political science problems)? If so, a good grade in real analysis would help your case, AND be valuable for your career.
 
Lastly, have you thought about taking some time off between college and grad school and working? Having your senior year grades available will probably help your case and a lot of people find it rewarding for other reasons. If you got a very good job in politics, it might also make you a better grad student.

Ya, you'll be fine.  People come from much weirder backgrounds than yours and you have a very competitive profile.  Just make sure to convey in your statement of purpose that you understand the types of research questions posed in political science. 
 
Also, you should add Stanford to your list.  Lots of good comparative work on Europe happening there. Scheve, Hainmueller, Rodden, Laitin etc.

Find a paper you really like, read the articles they cite.  Then go on google scholar and find out what the most-cited papers are that cite the new paper you found. Repeat.  Pretty soon, you'll have encountered (and possibly read), the most important stuff.  Take note of who the authors are and where they are teaching. You want to study with those people.

Most of the programs you're applying to have some version of a math camp, where they'll go over all of the minimal math you need in grad school. It won't do it in much depth, and learning a bunch of math in 2 weeks isn't the best way to retain it.
 
In particular, if you have any interest in quantitative research, you'll get a lot out of a linear algebra course.  Also, some statistics courses will help you understand the literature better.  If you want to be a methodologist, you basically can't take too many stats courses.  If you're doing formal theory (i.e. game theory), you would benefit from real analysis.
 
Have you taken any econ? In many respects, economics is the language of political science.  Taking intermediate would be valuable, especially at NYU and Stanford, which have many people trained as economists on the faculty.  
 
Being a part-time student won't be a problem if you can show that you used your time well.  But if you don't have much to show for it, then it could be an issue, particularly if  other applicants will have more to show from there time in college.
 
Have you asked your "distinguished" professors about these things?  What do they say?

Most of the programs you're applying to have some version of a math camp, where they'll go over all of the minimal math you need in grad school. It won't do it in much depth, and learning a bunch of math in 2 weeks isn't the best way to retain it.
 
In particular, if you have any interest in quantitative research, you'll get a lot out of a linear algebra course.  Also, some statistics courses will help you understand the literature better.  If you want to be a methodologist, you basically can't take too many stats courses.  If you're doing formal theory (i.e. game theory), you would benefit from real analysis.
 
Have you taken any econ? In many respects, economics is the language of political science.  Taking intermediate would be valuable, especially at NYU and Stanford, which have many people trained as economists on the faculty.  
 
Being a part-time student won't be a problem if you can show that you used your time well.  But if you don't have much to show for it, then it could be an issue, particularly if  other applicants will have more to show from there time in college.
 
Have you asked your "distinguished" professors about these things?  What do they say?

This troll was impressive.

This troll was impressive.

I doubt any good program would take you.  Econ PhDs regularly get jobs in political science departments, so they'd be suspicious of your need for more training. They'd assume that since you struck out in econ and couldn't get a faculty job in a political science with your econ PhD, that several more years of doctoral training wouldn't change things.
 
If you really want to learn more about politics, find some syllabi for PhD courses and go through the readings yourself.  Publish some papers in political science journals, and go on the political science job market.

This troll was impressive.

This troll was impressive.

Two things to bear in mind: 1) Stanford's placements page is missing a bunch of recent placements.  2) Stanford has much smaller grad student cohorts.
 
Some discussion on the first point here:
 
http://www.poliscirumors.com/topic/stanfords-placement-in-2013-2014-was-horrible/page/2

This troll was impressive.

This troll was impressive.

Congratulations! Many of you have the opportunity to become political science PhD students in the fall. Now that most of you have gotten in somewhere, though, it is necessary to examine whether going to grad school is right for you.
 
The reality is harsh and won’t be something you hear about at the recruitment weekends or from the DGS that spends his time here. At the best departments, only about half of the students finish with PhDs. It is far fewer as we go down the ranking tree. Of those, maybe half will end up with tenure track jobs somewhere. Of those, a little over half will receive tenure. Only one in eight members of your cohort will be in the discipline. Think about that for a little bit.
 
If you’re not that one person in eight, you’re wasting your time. It’s as simple as that. The skills you acquire in grad school have negligible applications to the professional world. Each year you spend at a PhD program is a year of work experience you don’t have and everyone who didn’t go to grad school does. Each year you miss will also have a new crop of college graduates enter the workforce for you to compete with. It’s not worth falling that far behind, especially when you know there’s an almost 90% chance you ending up right back on the non-academic job market.
 
So, you’re asking yourself, “why would faculty be so welcoming and inviting to me as we get closer to April 15? Don’t they realize that they’re likely sending me down a path to failure?” Well, guess what. They know and they don’t care. You are cheap labor to them, at best. Research assistants, teaching assistants and co-authors who do all the work on a paper (if you’re lucky). They need you. But that’s not even the half of it. Some faculty just enjoy recruitment for the sake of competing with old friends or rivals at other departments. Not only do they not care about you, their promises will literally disappear from your life as soon as you accept. That’s the world you’re entering. Be very wary. I know I wish I had been. 

It happens. If you have other offers from roughly comparable or higher ranked schools, you should try it. Worst case scenario is that they will say no. There is no shame in accepting the offer if they say no, and no shame in rejecting it if they say yes. As long as you do what you do in good faith, you need to pursue what is best for you. Grad Directors know that, and since they also want to get the best students they can, they will help you if it is in their power. In a sense, negotiating is actually mutually beneficial. 
 
You just need to remember that it might just not be possible for the DGS to make adjustments on the offer, so don't take it personal if negotiating gets you nowhere. 

Most successful political scientists do sustained work on one or a few related topics over their careers.  You listing a lot of weakly related topics is just setting off a lot of alarm bells, since most successful graduate students and faculty members do not and cannot sustain research in such a wide range of areas.
 
It's hard to say whether you should get a PhD and where you should do it without a particular area of current research in mind.  Political Psychology and Political Theory are both huge research areas, and schools will have many people doing different strands of work in those areas.  The other topics are also areas of research in their own right, though it's unlikely that any program would be able to serve you well in all of them.  
 
I'd suggest picking out some research in the areas you're interested in recent top journals (APSR, AJPS, JOP, WP, IO) or top university presses and seeing who wrote them and who they're responding to.  That'll help give you some guidance on who's working on those subjects.
 
If you are a plausible candidate for a top-6 school, you should also look through their faculties and pick out 3 people that you'd be happy working with at each place.  If you can't find 3 such people, don't bother applying to those places and go looking further down the list.  In general, you want to go to a top program or the place where you'd have the best advisor.  Start from the top and work your way down the list.

Most successful political scientists do sustained work on one or a few related topics over their careers.  You listing a lot of weakly related topics is just setting off a lot of alarm bells, since most successful graduate students and faculty members do not and cannot sustain research in such a wide range of areas.
 
It's hard to say whether you should get a PhD and where you should do it without a particular area of current research in mind.  Political Psychology and Political Theory are both huge research areas, and schools will have many people doing different strands of work in those areas.  The other topics are also areas of research in their own right, though it's unlikely that any program would be able to serve you well in all of them.  
 
I'd suggest picking out some research in the areas you're interested in recent top journals (APSR, AJPS, JOP, WP, IO) or top university presses and seeing who wrote them and who they're responding to.  That'll help give you some guidance on who's working on those subjects.
 
If you are a plausible candidate for a top-6 school, you should also look through their faculties and pick out 3 people that you'd be happy working with at each place.  If you can't find 3 such people, don't bother applying to those places and go looking further down the list.  In general, you want to go to a top program or the place where you'd have the best advisor.  Start from the top and work your way down the list.

No this is not what is happening. People aren't viewing you as someone who doesn't know what political science research is like because of your JD. Its that your research interests are overly broad as to be vacuous. Its that you are unsure what subfield you want to study. Anyway, American Quant said it better than me and with much more grace, so do what they say.

Most successful political scientists do sustained work on one or a few related topics over their careers.  You listing a lot of weakly related topics is just setting off a lot of alarm bells, since most successful graduate students and faculty members do not and cannot sustain research in such a wide range of areas.
 
It's hard to say whether you should get a PhD and where you should do it without a particular area of current research in mind.  Political Psychology and Political Theory are both huge research areas, and schools will have many people doing different strands of work in those areas.  The other topics are also areas of research in their own right, though it's unlikely that any program would be able to serve you well in all of them.  
 
I'd suggest picking out some research in the areas you're interested in recent top journals (APSR, AJPS, JOP, WP, IO) or top university presses and seeing who wrote them and who they're responding to.  That'll help give you some guidance on who's working on those subjects.
 
If you are a plausible candidate for a top-6 school, you should also look through their faculties and pick out 3 people that you'd be happy working with at each place.  If you can't find 3 such people, don't bother applying to those places and go looking further down the list.  In general, you want to go to a top program or the place where you'd have the best advisor.  Start from the top and work your way down the list.

Most successful political scientists do sustained work on one or a few related topics over their careers.  You listing a lot of weakly related topics is just setting off a lot of alarm bells, since most successful graduate students and faculty members do not and cannot sustain research in such a wide range of areas.
 
It's hard to say whether you should get a PhD and where you should do it without a particular area of current research in mind.  Political Psychology and Political Theory are both huge research areas, and schools will have many people doing different strands of work in those areas.  The other topics are also areas of research in their own right, though it's unlikely that any program would be able to serve you well in all of them.  
 
I'd suggest picking out some research in the areas you're interested in recent top journals (APSR, AJPS, JOP, WP, IO) or top university presses and seeing who wrote them and who they're responding to.  That'll help give you some guidance on who's working on those subjects.
 
If you are a plausible candidate for a top-6 school, you should also look through their faculties and pick out 3 people that you'd be happy working with at each place.  If you can't find 3 such people, don't bother applying to those places and go looking further down the list.  In general, you want to go to a top program or the place where you'd have the best advisor.  Start from the top and work your way down the list.

More thread hijacking...
 
Calculus was uses extensively in our first year program, in both our (required) stat methods and (not required) formal theory courses.  Differentiation was used to find maxima of MLE's, derive the information matrix, derive an optimal move in a game etc.  Integration to do expectations of random variables, derive conditional distributions etc.  There was also some very basic linear algebra: adding and multiplying matrices, inverting matrices, checking whether matrix was full rank, etc.
 
Calculus and linear algebra (and other topics) were taught in a math camp, so everyone was more or less brought up to speed, but that treatment definitely wasn't as good as having taken a full course in high school or college.  

Your GRE will likely kill any chance you have. How many times did you take them? the quantitative GRE doesnt really test math knowledge since it is all high school math.
 
The good news is that you can fix the gre score