Already in grad school, but is grad school for me?
On the other hand, I'm somehow having a hard time even figuring out who I report to, or if I have a me-project, or how to answer my questions about things like reporting and project ownership.


This seems like the real problem. If you don't have a well-defined goal, it's natural that you would feel like you're floundering. Advisors are busy; if you don't tell them you need guidance, they will assume everything is working out (until they realize that it isn't, at which point they'll blame you). So, I would recommend asking for a meeting with your advisor (or whoever is in charge of your lab) and try to better define your research goals.

I think everything else you describe comes from this. Once you have a good project and are getting interesting results, you will feel more like a part of the lab and some of the other angst will disappear. Your lab might never be a perfect match for you, but I suspect it will suffice in terms of giving you a PhD and a chance to figure out what's important to you when you choose your next position.


  Maybe it's as simple as "you're fine; this is normal. Get some sleep and go join a club to find friends. Then you'll get less unhappy, and your work situation will improve." I just don't know.


Pretty much. I would be aggressive in terms of finding a tractable project as described above. But I suspect the lack of sleep and friends are exaggerating the issue, and this is indeed very normal (as you say, there are dozens of posts here about it). 


  On top of it all, I'm no longer very sure that I believe in my subfield. It's not what I want to think about anymore when I go home.


This is perhaps the secondary problem -- after you get into a rhythm in your current lab/subfield, you may be able to pivot your projects towards areas you are more interested in.