When I was in college (late '60s) friends of mine would use the expressions, "Say what?" and "Do what?" usually when they had missed something during a conversation. I always assumed "Say what?" simply meant "What did you say?" and it seemed that "Do what?" referred to something that the speaker had described being done or wanting to do. Are they just variations on the same thing? Most of the friends using these expressions were African-American students in California.
Recently, I've heard it as "Do What Now?" of course as one big long word. This is in upstate NY.
I grew up in California, and say what? was always an expression of surprise or disbelief, not a request to repeat something that was missed. I would say that I got 100% on a test or ate a live worm and a common response would be, "Say what?" I never heard do what? but I suppose that would work just as well for the worm thing.
I grew up in southwestern NY and we used "say what?" exactly the same way. It also is used to ask someone to repeat what they have said but not as much as a statement of disbelief.