So I'm in the process of building some custom panels and stuff, not big things, but enough that I will run out of USB ports eventually. I'm pretty sure you guys have run into this too and I guess you use hubs for that, right?

I know next to nothing about USB hubs... All I know is that I want one with a power cord. I get that, you need the amperage, but then what? Are there any caveats when it comes to joystick-like things? Anything in particular I should watch for? Can I mix USB 2.0 and 3.0 here?

What I imagine I'll end up connecting are standard USB controllers, nothing fancy, some Teensy here, some gutted game pad there, maybe some Arduino over here (both "in" and "out" comms, though).

If you have devices that plug into it that are USB 3.0 the hub itself might need to be 3.0 as well other than that USB has always been backwards compatible. And aside from the point mentioned about needing a powered hub (a must with game devices) I think you're on the right track. Just check out reviews on amazon or something before you make a purchase to make sure whatever you get doesn't have some glaring issue.