I'm a little confused on the difference between "maken" and "doen". Is there a clear distinction, or do I just need to memorize when to use each verb? Also, could "Ik wil geen huiswerk maken!" be used in a hypothetical situation where a professor does not want to make homework for his/her students?
There is a clear difference in Dutch (unlike in French or German) and they closely resemble the English words "make" (create or fix something) and "do" (perform an action). This sentence happens to be one of the few expressions where English and Dutch use a different noun: in Dutch, you "make" your homework, instead of doing it.
And yes, "huiswerk maken" could technically also refer to a teacher creating a problem set.
"Maken" is when you create something, like making a scultpure: "Ik maak een standbeeld." "Doen" is when you do something, like walking. About the professor: yes, it can be said, but "Ik wil jullie geen huiswerk geven!" ("I do not want to give you homework!"), is much more likely and logical. I hope this answers your question.
Also, I think the rule is (and if it's not completely correct, or just incomplete, let somebody correct me please :) ): when any one of kunnen, willen, blijven, gaan, zullen, mogen, moeten is used as a modal verb, and in combination with an infinitive verb, then "te" mustn't be added.
You should definitely fight that urge ;) in this sentence "te [...]" is wrong