Obviously I am not in the admission committee so I could be wrong, but the following are what I see and what I hear from others.
1. Yes. GRE mathematics is more important than GRE. In terms of the schools that you want to apply to, you should score at least 800 (around 80 percentile). Nevertheless, for less competitive schools, if the rest of your application is strong enough, you could get in with a score around 700 (around 60 percentile).
2. You should have taken both of the tests by October 2019 if you plan to apply for Fall 2020 programs.
3. You could search for REU programs offered by the schools that you are interested in. You could also take some classes from the local schools you are interested in.
As for your number theory and topology interest:
2. If you want to take topology, you should know some basic algebra, such as tensor products, resolutions, etc. If time permits, you can read Hatcher's "Algebraic Topology". You should pay some attention once your class starts to talk about homology/cohomology, which are important in many algebraic areas. However I think most topology classes would teach in a topological fashion so that the homological algebra you learn in topology may be fairly different from the homological algebra you will need in algebra. Again, I don't have any background in topology except taking the topology qual class (which is approximately what Hatcher covers in his book), so I cannot give you any good advice. You can probably consult some topology professors in your department. You can also do some reading courses with them if you are interested.
3. I talked to one of my professor, who worked in graduate admission for five consecutive years. He told me that many PhD programs do not expect you to have any interesting result if you are conducting research in theoretical mathematics. However, it is better if you try to do some research, for example, in REU programs.
Last edited by ArtinWedderburn
on Tue Jan 09, 2018 12:07 pm, edited 1 time in total.