I’m sure citing online sources such as a posts on /r/AskHistorians will be acceptable in my essay, especially given how well this sub is moderated and given that I'm not a history major.

As university faculty, I would recommend that you do not assume this at all. We in AH are not infallible, or entirely peer reviewed. There is a great deal of rigor, but there is also a huge amount of anonymity and sometimes things slip by even the most diligent gatekeepers. Besides, this is a moving target. Unless your subject is this subreddit itself, you should be skeptical of using posts here as authoritative statements in a paper absent any independent verification by you. I mark people down harshly for making this assumption, and indicate in my syllabus that online sources need to be handled with care and consultation.

The other comment is thus quite correct that a good and valid answer on this sub will have pointers to sources; according to The Chicago Manual of Style it is always best to go to the original source and avoid what we call "secondary citations" (X, quoted in Y) unless that is absolutely unavoidable. There are formats in most major style guides now to cover blog posts, forum posts, and so on; find out what style guide your instructor prefers and investigate its prescriptions. At the very least, if you can't find any clear answer, talk to your instructor. When dealing with citations, I am always more than happy to talk to students who want to handle it correctly, and I'd imagine others would be too.

Out of curiosity, which posts are you looking to cite?

Awesome that you're striving to give proper credit!

For the author name you'll have to use the nickname, unless it's somehow publicly known who's behind the nickname. The title of the work would be the question (or "RE:" the question). And of course include the full permalink to the comment.

My APA handbook gives the following example:

Middlekid (2007, January 22). Re: The unfortunate prerequisites and consequences of partitioning your mind [Web log message]. Retrieved from http://.....

(With the "http://...." of course being the full url.) Other style manuals might have different templates.

Quoting parts of an answer with proper attribution shouldn't be a problem. Quoting a full answer might run into copyright issues (these laws - and their exceptions - vary per country/region). For posts on /r/AskHistorians copyright in principle remains with the author, so any (potential) issues should be resolved with them.