Use of owaru

終わる is to end, finish, or close.

I know it can be used in the same sense that "terminate" can, but can it be used as a verb of finishing doing something?

ex. もう終わったあのしゅくだい？

If not, what verb would work here? すむ ?

Yes, you can say that (the word order needs to be changed a bit, though what you wrote is OK as completely colloquial speech):

もう あの宿題、終わった？ (informal speech)(Is that homework done/finished?)
もう あの宿題は 終わりましたか。(polite speech)
終わる・終わります means "(something) finishes."

You could also use other verbs, like 終わらせる・終わらせます, which means "(you) make something finish," so "(you) finish/end something" and 終える・終えます which means "(you) finish/end something."
もう　あの宿題、終わらせた？ (Did you get done with that homework? - informal)
もう　あの宿題は　終わらせましたか。(polite speech)


済む、すむ also means to finish, and it has the connotation of "(something) gets done, as it should."
You can say:
もう　あの宿題、すませた？(Have you gotten done with that homework? - informal)
もう　あの宿題、すんだ？(Are you done with that homework/is that homework all done with? - informal)

Perfect. Thank you so much. I hadn't thought of using causative form.