1. I know you stated that you are not necessarily looking for feedback on whether this is even a worthwhile endeavor for a JD student, but answering the question "why do I want a PhD" is a necessary question to answer before any other questions can be answered regarding your chances for success in being accepted into a grad program. To be sure, it is important that you discuss this with your professors to get their feedback, but keep in mind they may be biased in their assessments of whether you should pursue a graduate degree (something I have learned in my research).
Basically there are only two reasons to get a PhD: 1) you are passionate about research and want to do so at institutions that will provide you with the best resources to accomplish your research goals; 2) You want to be a teacher at a research oriented institution. Any other reasons--wanting to advance in your career, not knowing what else to do with your life, or thinking getting a PhD will get you access to higher paying jobs--are bad reasons for wanting to get a PhD. This is because a PhD program is meant to do one thing: socialize you into thinking like a researcher and providing you with the tools to become successful at a research oriented university. Consequently, the skill set you obtain in a PhD program is rather limited and is not always applicable to jobs in the private sector. This is not to say you couldn't get other jobs in government or at think tanks with a PhD--indeed many grad programs are now encouraging students to look for jobs outside of academia (more on that later). But the opportunity costs of getting a PhD (which on average is 7 years for poli sci programs) is high. This leads into the next reason why you should really consider whether getting a PhD is the right thing for your professional career goals. The job market is absolutely atrocious. There are significantly more PhD graduates than there are positions available, which has created a whole class of students who are unemployed or perpetually working as adjuncts with very minimal pay.
If you are interested in working at a think tank or public policy more generally, you could still get a job with just your JD. I know from personal experience that many think tanks accept JD graduates in research roles because I am currently working at a think tank right now, and when I was looking for jobs at think tanks, many job listings listed having a JD as preferable. So if this is your goal, just applying to think tanks or government research programs out of law school may be a smarter choice due to the reasons I mentioned above. If you really feel like you want more of a public policy oriented education in addition to your JD, then getting an MPP or MPA is probably a better option. Here are some links that will provide more information on what I have been discussing: https://chrisblattman.com/about/contact/gradschool/ http://100rsns.blogspot.com/
2. Now to answer your questions and give general feedback.
I have spoken to current poli sci grad students, including one who got his JD and two others from Stanford. They have all said that schools are indifferent regarding whether you graduated with a JD. Schools mostly care about your potential for producing interesting, original research and becoming a successful researcher at a highly ranked academic institution. This is due to the fact that schools are risk averse, and don't want to expend large amounts of faculty and monetary resources on students who have no chance of graduating with their PhD or getting hired as an academic. So you need to demonstrate that you have interesting questions and that you have potential for producing exceptional research.
My guess is, it probably won't make much of a difference, due to the goals of law school and PhD programs are different. A law school's goal is to train students to think like lawyers and equip them with the proper tools to become successful lawyers. This is fundamentally different than the goals for PhD programs, which is to train you to become researchers. One has a more practical focus, the other has a more theoretical focus. Consequently, going to Yale Law School won't signal much to admissions committees your potential for research, nor will it provide an accurate barometer of the rigorousness of the program in relation to what is expected of students in a PhD program (again being due to the fact that what is expected in law school and PhD programs are totally different).
It might help, especially if you publish articles about international law as it relates to international relations, and can get those articles published in political science journals. Moreover, there are political scientists who publish in law journals. But I am not sure if publishing in a law journal would signal anything to admissions committees, since the standards of what gets published in political science journals and law journals is likely different. It certainly couldn't hurt, but I would definitely consult someone on that.
I don't think it would hurt necessarily, but not having research will not give admissions committees an indication for your competency in research, or your potential for success in research. Perhaps if you publish in a political science or law journal, it may help overcome your deficiency in research experience.
3. Here are a couple of other notes regarding your profile.
a) the fact that you have a 4.0 in undergrad is excellent. Although it doesn't necessarily say anything about your ability to do research, it will help you get past the first round of cuts admissions committees make.
b)the fact that you don't have a huge quant background won't hurt you. Political Science is not economics, which requires that you have a quant heavy background to be considered in top programs. From speaking with other current poli sci grad students, ad coms understand that most people applying to their programs probably don't have a heavy quant background and thus compensate for this fact by requiring you enroll in quant classes in your first year of a grad program. However, this is why it is crucial that you do well in the quant section of the GRE, because it signal to ad coms your success in the required quant classes. If you want a career as a professor at a research oriented school, you need to get accepted into a top 20 program. This means you need to get at least a 160Q score, but preferable a 165 or above.
c) it is critical that you get letters of rec from professors who know you very well, and that you know will ham it up for you in their letters. According to one grad student I spoke with, it needs to be on the level of "this the best student I've had in the past 10 years", or something to that effect. It would be preferable to get most or all of your letters of rec from a poli sci professor (like the one you mentioned that was your advisor on your senior thesis), but if this is not possible, then get the other two letters from law school profs who are either well known in the poli sci circles (i'm sure there are a few at yale) or are just well known in the legal field generally, AND who know you and your work very well. Again, letters of rec from poli sci profs are preferable because poli sci faculty in ad coms will share with your poli sci letter writer a similar gauge of what success in a student looks like than would someone who is a faculty member from a law school. Consequently, ad coms place greater weight on letters of rec written from poli sci profs than from law school profs.
Overall, in order for you to be competitive at a top school, you generally need a high GPA, high GRE scores, an exceptionally well written statement of purpose indicating your research interests, faculty you would like to work with, and what interesting research questions or ideas you will bring to the table, and glowing letters of rec. Publications would also be a nice resume booster. Of course, none of this guarantees you will get into a top program, because acceptance decisions are fairly random; it will merely make you competitive with other applicants.
I know this is a lot of info to digest, but I hope it helps. If you have any other questions, feel free to message me. Ultimately, I highly encourage you do more research by speaking to current law school profs, past poli sci profs, and reaching out to current poli sci grad school students.