Look I have the utmost compassion for your boyfriend. I am an alcoholic in recovery with 9 years sober. But this situation sounds fucking insane. No offense OP, I'm going to be honest though. For starters your boyfriend left for treatment and didn't say anything and kinda left the relationship off on a note that it was over. I don't think you are at fault for carrying on and feeling like you were single after that.

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But you hooking up with the neighbor is a big big nono. You never shit where you eat. Obviously you're an adult living on your own, if you want to hook up with your neighbor you can. But at least for me there are always a ton of issues with this in my experience. For example there is a cute neighbor girl who was giving me the side eye a few months ago. I was single at the time and could have hooked up with her if I wanted to. But what if she and I hooked up and then had a messy breakup? She knows where I live and can key my car for example. Also let's say I breakup with her and I continue to date, she may try and talk shit to or confront another woman I'm dating as she's coming into my building or leaving my building (like what happened to you). Incidentally I ended up dating other women and found and am with someone who is my current girlfriend and I've seen neighbor girl give the dagger eyes when she saw us too together one time. I can only imagine if I had some kind of brief relationship with neighbor girl she'd probably do a lot more than give dirty looks.

And in your situation you came on a worst case scenario to this, as now you have to move in order to try and get things back to normal. My advice is to try to explain to your boyfriend why you did what you did. That he ended things very ambiguously and that you thought you were single. What you did was not cheating, and he's not gonna like hearing it but it's true. I hope you move and you can work through this with your boyfriend OP. I wish you the best of luck.