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    Good linear algebra book recommendation

    If you want a rigorous treatment of the subject, use this book:
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    When I took it I used the Princeton Review's book to prepare from. Certainly do as many practice exams from here to the test date, and make sure you understand every problem you get wrong to the smallest detail. Good luck!
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    Schools Can I skip College Algebra?

    Yes. Speaking from experience, I taught myself calculus without really having a strong background in trig nor algebra beforehand. Naturally you won't be able to fly through the material as you'll have to reference some things, but after that it is smooth sailing. I personally feel in retrospect...
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    Programs Will my past keep me from getting into a top PhD program?

    If you apply to graduate school for Physics they will look at your coursework in Physics and your potential to do research in Physics. So PoliSci does not matter.
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    So many mathematics choices

    If you are planning on going to graduate school, then settle down for a postponement of Graph Theory until then. Or even better, check out two books on graph theory and teach yourself from them! But do not prolong on taking Real Analysis as that course is essential, and it is better you take it now.