Aleph-one

I'm a second-year math undergrad at Northwestern. I have a few questions, thanks for helping!

1) I plan on taking the Math GRE subject test during my senior year. By that point I will have taken real analysis, algebra, topology, number theory, probability, principles of higher math (set theory, logic, cardinality, functions), and upper-level linear algebra. We don't really offer a standard complex analysis course for undergrads, so I'm planning on independent studying it with a professor during my senior year. Is complex analysis on the GRE and how much is necessary to know? What about DiffEq? My advisor told me not to take it...will this hurt me?

2) What's the best way to get involved with research? I plan on doing an honors thesis during my senior year, but I want to get involved sooner. I'm hoping to do something over the summer at my home in Los Angeles, but people at Caltech seem uninterested in working with non-Caltech students. I will be applying for a program at UCLA, but it definitely seems to have an applied edge to it. Is it possible to work on my own on pure math research? Where can I find unsolved problems that are somewhere near my level of abilities? I won't be taking analysis, algebra and topology until next year...but I do know how to write proofs and I have no problem with studying math on my own time.

Okay yeah I'm sounding like an over worried, nerdy undergrad loser.....but I hope you guys can help. Thanks so much!