REU deadlines are probably past due too, but you could check. I got into a couple REUs in early March many years ago. A couple rejections in February. Withdrew my other apps after I chose my program, so no idea how late the other apps would've taken to hear back. Still, most of the deadlines are probably past, but you can check.
To be honest, without knowing anything about you, it sounds like you're more interested/focused on coursework than research. You list a million courses and talk about filling in gaps in your knowledge. If you want to take all those classes, perhaps you should do a Master's. It's very possible to do a PhD without ever studying measure theory, functional analysis, complex variables, differential equations, etc. PhD is about research, and the further you go, you'll realize the more you learn, the less you know.
Edit: Next summer? Probably no point. You'll either have a grad school acceptance or you won't. You probably won't be eligible if you'll be a graduating senior anyway.