Scholarships are dished out all the time. If you're a good candidate for top 20 schools, then schools ranked below 20 will offer you pretty decent money.
Take the GMAT as soon as you feel completely prepared. If you're shooting for Round 1 then it would be good to be done with the GMAT by July or August.
I applied Round 2, took the GMAT the very end of October which gave me two months for essays and applications which was plenty of time.
Just make sure you don't burn out studying and say to yourself, "Screw it, I'm just going to take the GMAT and see what I get, and if I don't do well I have plenty of time to retake." You should really aim to take the test just once. As soon as you are achieving a test score on GMAT Prep that is around what you desire on the real thing, you're ready to test.