Hey, let me try to answer this.
1) You get a professor there to agree accepting you as a Ph.D. student during PSI, and .... that's pretty much it.
2) Read above? I guess you'd need decent/passing grades in most courses. I've heard Freddy is quite selective with his students.
3) If you want to go do Mathematical Physics with Kevin Costello in the same year many other students want to do the same thing... then I have some bad news for you, there's going to be serious competition, I'd wager. I think it's easier to work with some profs than with others (less competition and more open to students)
Remember that the first step for a Ph.D. in PI is to go to PSI, and that they select ~30 students for PSI each year. Don't be discouraged if you aren't selected.