Okay, part of this mystery is solved-- The “Stacks (0)” and nothing listed in the Lambda page is because I’m apparently working on two different servers, which I suppose is pretty similar to what you said, though they’re not two different accounts. I’m not 100% sure how that happened, but noticing the droplist in the upper-right corner just resolved a lot of confusion for me. (I never expected that.)
I still haven’t got the C# working, but my superiors informed me that they’d rather do Node.js instead. I’ll leave this here in case anybody else wants to comment for people in the future coming by this post.
Update: And I just realized that’s why nothing was listed in the API Gateway page, either, so, I suppose this is solved.
Update 2: I forgot to say thanks to Moro. Thanks, Moro!