I would advise that you read Not Even Wrong or The Trouble with Physics to see that string theory research isn't quite the glamorous sparkling wonderland Brian Greene, Michio Kaku, et al make it out to be. I never really understood the fascination with string theory; even when I was just...
I think a lot of the research in string theory is shifting from trying to use it as a GUT to using it as a framework for working out new mathematical methods. For example, some methods originally developed in string theory have been used in condensed matter contexts.
I would contact the appropriate representative of DAMTP and let them know the details of your educational background. They would then know if it would be appropriate for you to enroll in Part III.
A First Course in Loop Quantum Gravity by Gambini and Pullin might be a good book for you. It says it's on LQG, but half the book deals with background material (SR, GR, QM, QFT). Also, it was specifically written for undergrads. It's a short relatively easy read, and not too expensive.
A...
I would say choosing the Nobel laureate just because they've won the Prize would have been a mistake. A lot of Nobel laureates have not been the student of a Nobel laureate and a lot of Nobel laureates supervise students who end up becoming nobodies.
Try using your academic/departmental advisor. The questions you're asking are what they are there for. They're much better suited to respond to your situation than us internet folk.
Spivak is probably overkill for what you want (it's a 5 volume series). Since it sounds like you're more interested in applications, try "The Geometry of Physics" by T. Frankel and Arnol'd's mechanics book (I forget exactly what it's called). These books introduce differential geometry and the...
"Better" is a highly subjective term. I used Arfken/Weber for a class and I really didn't like it. It's more of a reference book than a learning book. The explanations are too brief properly learn the material.
It doesn't assume you know calculus at the start, but it doesn't avoid calculus either. It has short interludes between chapters that introduce mathematical concepts such as calculus. It introduces the mathematics as it needs to. Similar to how most physics students learn their math in the...
The Book's actually been out for a little bit (I saw it 2 weeks ago at B&N). It essentially covers the basics of classical mechanics up to the principle of least action and a little bit of electromagnetism. It is intended to be able to be read by someone without a good knowledge of calculus. I...
You mentioned GR as one of your interests. A lot of the theoretical work in GR is labeled as "Numerical Relativity." It is highly dependent on computation. In string theory &c., you can't calculate anything, so nobody does. Also, string theorists seem very passionate about having a theory which...
A lot (most?) of theory work is computational. It's hard to tell if your theory matches experiment without running some simulations.
For analytical math type theory work, do you know any differential geometry? It's essentially a pre-req for what you want to do, plus it has applications to...
Why do you want to make $150k+ per year? Are you really going to be spending that much money? Chances are, you'll probably be just as happy making half of that (and there's psychology studies that can back this up).
It probably works well if you already know/knew the material. From experience, I don't recommend trying to learn the techniques for the first time with this book. The explanations are too brief.
I don't think it has to be an academic organization. Do you work for a company? I've seen postings on arXiv by people working in industry. Even if you are self-employed, that should be fine. If you are unemployed, your work will probably be viewed with a high amount of skepticism.