How much will the GI Bill cover before I have to pay out of pocket?

Honestly, the best answer is "it depends." Your best bet is to look for a school that participates in the Yellow Ribbon program. These institutions often match whatever is paid for by your GI Bill, and many find it can cover most if not all of the tuition, at least. The people I know who have used the program did so for specialized degrees like a JD or Doctor of Nursing, and also qualified for in state tuition rates. I am not sure of the exact requirements for your degree, which may incur extra costs, but your best bet would be to start by finding the best school that offers what you want and also works with benefit matching.

Unfortunately, I don't have the article right now, but I recently read about a push for all members using TA or GI Bill benefits to fall under in state tuition rates. This would also be something great for you to inquire about as you research your options.

While the GI Bill will not carry you all the way through to your PhD, it will cover a lot of your undergraduate courses. I would seek out and ask your school's Veterans' Rep for a better understanding based on the school you wish to attend.

It covers 36 months if you are using the post-9/11. That's equal to 4 school years, if you do summer school it's closer to 3 years. So it depends on how long it takes you to get your Ph.D. As far as I'm aware the limit on the number of classes is determined by the university.