gaillardsghost

Straight from Wikipedia, and relevant to several things you wrote:
 
Relationship with Columbia University[edit] Teachers College holds its own corporate status separate from Columbia University, including an independent board of trustees, budget, endowment, and multiple admissions standards; however, as noted in Columbia University's Faculty Handbook, the College is an official faculty of the University, serving as the University's Department of Education.[9] Teachers College faculty hold Columbia University appointments; its President is a dean of the University; and all students receive their degrees from Columbia University. As with the other Faculties of the University, Teachers College Ph.D. degrees are specifically conferred by Columbia University's Graduate School of Arts and Sciences.[10][11]

^ Second the above.  While you already seem like a strong candidate (previous publications, internships that are relevant), I think your SOP will be your strongest card in the game to proving that you have the maturity and tenacity to make it through a PhD program. For example, your SOP should probably explain why a PhD is a better fit for you, rather than a MA or MEd. What is your end goal w/a PhD? Why is it an essential?  That's probably the golden question to answer there. :)

For what it's worth, I worked with someone who graduated with a PhD from Teachers College. She always used to say, "A degree from Columbia takes you places.." I've never heard anyone say, "Oh, it's just Teachers College." Whether it is formally or completely linked with Columbia University or not, Teachers College has had a 100+ year presence in the education world, and certainly has a reputation that far exceeds most other graduate schools of education--I think that's the thing to keep in mind more than the 'Columbia connection' or relationship.

Straight from Wikipedia, and relevant to several things you wrote:
 
Relationship with Columbia University[edit] Teachers College holds its own corporate status separate from Columbia University, including an independent board of trustees, budget, endowment, and multiple admissions standards; however, as noted in Columbia University's Faculty Handbook, the College is an official faculty of the University, serving as the University's Department of Education.[9] Teachers College faculty hold Columbia University appointments; its President is a dean of the University; and all students receive their degrees from Columbia University. As with the other Faculties of the University, Teachers College Ph.D. degrees are specifically conferred by Columbia University's Graduate School of Arts and Sciences.[10][11]

Straight from Wikipedia, and relevant to several things you wrote:
 
Relationship with Columbia University[edit] Teachers College holds its own corporate status separate from Columbia University, including an independent board of trustees, budget, endowment, and multiple admissions standards; however, as noted in Columbia University's Faculty Handbook, the College is an official faculty of the University, serving as the University's Department of Education.[9] Teachers College faculty hold Columbia University appointments; its President is a dean of the University; and all students receive their degrees from Columbia University. As with the other Faculties of the University, Teachers College Ph.D. degrees are specifically conferred by Columbia University's Graduate School of Arts and Sciences.[10][11]

For what it's worth, I worked with someone who graduated with a PhD from Teachers College. She always used to say, "A degree from Columbia takes you places.." I've never heard anyone say, "Oh, it's just Teachers College." Whether it is formally or completely linked with Columbia University or not, Teachers College has had a 100+ year presence in the education world, and certainly has a reputation that far exceeds most other graduate schools of education--I think that's the thing to keep in mind more than the 'Columbia connection' or relationship.

The affiliation between TC and Columbia is complicated and is at least in part an artifact of political and legal and historical quirks that no one cares about at this point.  It is an arrangement that highly unique in post-secondary education.
 
They are the ed school affiliated with Columbia.  They will always be the ed school affiliated with Columbia.  "Teachers College, Columbia University" is the accepted convention as to how to represent the name of the institution.
 
As to hiring committees: opinions about the quality of TC vary widely, but I would argue that has little to do with TC's relationship to Columbia.  There are other schools in the Columbia universe that have equally up-and-down reps (SIPA, Arts)...regardless of the legal structures of their relationships.

If you are interested in Gifted Education, I recommend reading Dr. James H. Borland out of Teacher's College, Columbia University. His research and recommendations have completely changed how I view and teach the gifted. Furthermore, Columbia has a master's degree for gifted education. I'm pretty sure Dr. Borland teaches some of those classes. I'm looking into it myself.
 
http://www.tc.columbia.edu/academics/index.htm?facid=jhb27
http://www.tc.columbia.edu/c%26t/gifteded/

About five years ago I unsuccessfully pursued gifted and talented positions. I guess I am still technically certified in G&T, though I haven't done anything with it recently. Take what I write here with that in mind--I don't know if things have changed, as I've been into other things.
 
Between five and seven years ago, the opportunities seemed few, and most of them where I was located were elementary based. There were secondary opportunities here and there, mostly in a large urban district that I taught outside of (this was out west, in the US). When I relocated to the east, I was able to get interviews for three different positions in two different states, neither of which I was licensed in at the time (and one of which I was a finalist for in a *great* place). I also had an interview in a district I was working in at the time. The lack of success here probably had to do with lack of experience, gifted specific coursework, and to a lesser extent the credential issue--at least, that's what I think.
 
As far as advice in pursuing this area:
 
1. Pursue certification in multiple areas (e.g., math/science, language arts/social studies, although other combinations may prove useful). A teacher of the gifted sometimes has to provide support across content areas, or at the very least advise content teachers in those areas about how to modify content and instruction.
 
2. At the secondary level, it seemed like many gifted and talented positions required English/language arts certification. I don't know if this was just the area I lived in, or if this is part of a larger trend. I saw a few that were math-focused out west as well.
 
3. Gifted endorsements or other teaching credentials may get you into interviews, although you'll need more obviously to win yourself a job. Such licenses often require graduate coursework in gifted and talented education. Some states, like New Mexico, require a content test as well or instead; they used to take the PRAXIS II in gifted, but states may have their own G&T test instead/as well.
 
4. Teach, teach, and teach! My impression from the districts I worked in was that they often staffed these positions internally, from people who had a fair amount of experience (i.e., 5 years or more). Not saying you won't find a position like this with less than 5 years of experience, but these districts I'm referring to seemed to think it was easier to find a new content teacher than to hire a gifted specialist from the outside.
 
5. Special education credentials are a double-edged sword when it comes to gifted. The skill sets of "special educators" and "gifted educators" have some overlap between them. States that use IEPs and gifted IEPs (are there states that don't?) require gifted educators to be familiar with legal protections and reporting that special educators regularly must use. However, in some districts, a license in special education means a possible reassignment to a special educator role, even if one is hired initially to teach gifted and talented. In terms of being hired, it should go without saying that special education credentials and experiences will land you a job much quicker than gifted credentials will.
 
Feel free to message if you have other questions. I'm not sure how much I'll be able to help, given the experiences I've had, but I can at least entertain questions and see if I can help.

No, I'm pretty sure the masters are conferred by TC (specifically the "trustees of Columbia University").  
Many years ago, TC used to be a freestanding school.  That's why it is semi-autonomous.  
I think of it as being analogous to Barnard.  Barnard is a liberal arts college in its own right, but it is also the women's college of Columbia University.  No matter the field, if one knows Columbia, one also knows Barnard and TC.  They're under the Columbia University umbrella.   

About five years ago I unsuccessfully pursued gifted and talented positions. I guess I am still technically certified in G&T, though I haven't done anything with it recently. Take what I write here with that in mind--I don't know if things have changed, as I've been into other things.
 
Between five and seven years ago, the opportunities seemed few, and most of them where I was located were elementary based. There were secondary opportunities here and there, mostly in a large urban district that I taught outside of (this was out west, in the US). When I relocated to the east, I was able to get interviews for three different positions in two different states, neither of which I was licensed in at the time (and one of which I was a finalist for in a *great* place). I also had an interview in a district I was working in at the time. The lack of success here probably had to do with lack of experience, gifted specific coursework, and to a lesser extent the credential issue--at least, that's what I think.
 
As far as advice in pursuing this area:
 
1. Pursue certification in multiple areas (e.g., math/science, language arts/social studies, although other combinations may prove useful). A teacher of the gifted sometimes has to provide support across content areas, or at the very least advise content teachers in those areas about how to modify content and instruction.
 
2. At the secondary level, it seemed like many gifted and talented positions required English/language arts certification. I don't know if this was just the area I lived in, or if this is part of a larger trend. I saw a few that were math-focused out west as well.
 
3. Gifted endorsements or other teaching credentials may get you into interviews, although you'll need more obviously to win yourself a job. Such licenses often require graduate coursework in gifted and talented education. Some states, like New Mexico, require a content test as well or instead; they used to take the PRAXIS II in gifted, but states may have their own G&T test instead/as well.
 
4. Teach, teach, and teach! My impression from the districts I worked in was that they often staffed these positions internally, from people who had a fair amount of experience (i.e., 5 years or more). Not saying you won't find a position like this with less than 5 years of experience, but these districts I'm referring to seemed to think it was easier to find a new content teacher than to hire a gifted specialist from the outside.
 
5. Special education credentials are a double-edged sword when it comes to gifted. The skill sets of "special educators" and "gifted educators" have some overlap between them. States that use IEPs and gifted IEPs (are there states that don't?) require gifted educators to be familiar with legal protections and reporting that special educators regularly must use. However, in some districts, a license in special education means a possible reassignment to a special educator role, even if one is hired initially to teach gifted and talented. In terms of being hired, it should go without saying that special education credentials and experiences will land you a job much quicker than gifted credentials will.
 
Feel free to message if you have other questions. I'm not sure how much I'll be able to help, given the experiences I've had, but I can at least entertain questions and see if I can help.