Do, Did or Have?

"Did you understand (what the lecturer said)" or "Do you understand (what Hilbert Space is)?"

Even during the class do you say "Did you undestand?"

If it's something that was just explained, and you want to know if your classmate understood what was just explained, then yes, you say, "Did you understand (that [explanation / point / concept / principle])?"

If you want to know in a more general way if someone understands, you say "Do you understand (this material)?"

In the typical usages, Do you understand (xyz)?

How about

(2)Have you understood?

To me, it works quite well in BrE.

It really doesn't seem too much used in AmE.

In AmE I don't think we'd use Have you understood? if any significant amount of time had passed since the moment when potential understanding might have taken place.

Immediately after explaining something to a student one might say in AmE Have you understood? (or Do you understand? or Did you understand?) But a few minutes later, after a short interru

Thanks, I understood.[You don't usually say "I have understood" in AmE, right?]

Thanks, I understand.

I'd say I have understood is quite rare in BrE too. An example could be I have always understood that A means the same as B, but now it seems they're quite different.
Or: How much have you understood so far?