I think it's a bit early to ask if you can become a physicist. If you like science, then pursue it. You don't come close enough to knowing enough physics/math to know whether or not you would like to major in it much less go to grad school.
Just relax. Work on improving your grades and maybe...
Great book. I think a (HS) student with a deep interest in mathematics would enjoy his informal, yet rigorous exposition to the material. If one already knows proofs, a more advanced book would probably be more appropriate.
What is the description for vector analysis. I would guess PDEs are useful for appplied, but I'm not qualified to say it's true nor what you should take. I'd suggest for someone else's opinion. Ask your professors; they know the most.
If you look at this webpage, you can see some of the Analysis problems. If they interest you, you might want to take it.
https://web.math.princeton.edu/~adeptrep/lowerdiv/215/MAT215
It's great if you can keep up with the classes, but I would think Real Analysis already assumes a good grasp/knowledge with proofs.
EDIT: Oh - it's baby Rudin. It's not that bad then. At Princeton, Baby Rudin is used as a honors math track. It's rigorous, but if you commit to it, you'll do...