Well, due to Zil'ber's theorem any totally categorical theory is not finitely axiomatizable. So yeah, so is your example :)
I think it shouldn't be hard to prove in this specific case (an idea using ultraproducts and Los' theorem comes to mind, should check if it works), but the general proof requires a fair share of some basic geometric stability theory. The only book I know which presents it is Pillay's, but unfortunately it isn't a very readable book.
I found this book below. Do you think it'd be an overkill to study it?
Deriving it in a geometry class would be like deriving any of the area formulas (which certainly has to be done). It'd be a formal proof just like everything else in the class. There might be algebra involved in that proof.
My experience was reading through a high school geometry text book and doing some of the exercises. That at least means that any students using the same book that I did won't have done any algorithm churning, since those sorts of exercises just weren't in the book.
It was this book. A typical high school geometry text as far as I'm aware.
At what level? Sutherland's Introduction is good. I also recommend Korner's lecture notes.
I enjoyed this book for a functional analysis view:
An introduction to nonlinear analysis by Schechter it goes into calculus of variations nicely.
Also, strogatz provides a nice introduction to dynamical systems.
Introduction to perturbation methods by Holmes is decent too if you want to see that, but it doesn't have too many pfs.
You might want to post in /r/math
1.) Complex Analysis is easier than Real Analysis. Everything that you would want to work there does. I'd recommend picking up a textbook and just going through it. "Visual Complex Analysis" is a good book to start with.
2a.) I assume you're going to be applying in the Fall. There isn't too much you can do now, but make sure you have a few solid connections with professors. Go to a few seminars or colloquiums and just be interested and get to know some, if you don't already. They don't even have to be in Algebra. Letters of recommendation go far. Look for people who can be honest, not necessarily people with an impressive title or CV.
2b.) To actually prepare for grad school: Read. If you're into Algebra, pick a series of books to go through. I'd recommend a progression like Basic Algebra -> Commutative Algebra -> Algebraic Geometry -> Linear Algebraic Groups. While reading these, have a few things on the side, you'll eventually need everything. Some things that are helpful are Algebraic Topology, Abstract Harmonic Analysis, Differential Geometry, Number Theory, I like this book on Homological Algebra. Don't read these for a complete understanding, just a familiarity.
Elliptic curves originate from calculus and there is an analytic theory for them (the Weierstrass P-functions and all that), but they are also very interesting from an algebro-geometric viewpoint, closer to the abstract algebra that you mention.
Fundamentally, the points on an elliptic curve form a group (an algebraic structure) under the chord-and-tangent method of addition. The (same) group law can also be defined via the Picard group of the curve. Have a look at the table of contents of Silverman's Arithmetic of Elliptic Curves to see what kind of algebra you can encounter in the theory of elliptic curves.
Barrett O'Neill teaches you all the math you need (semi-Riemannian geometry) and then gets into relativity. It's one of my favourite texts, and it's very readable as an introduction to the subject.