Question: I'm trying not to sound like a...

Get. Out. NOW.

Red flag one. he mixed your name up with hers, red flag two. he takes interest in her apparent 'dislike of rain' rather than whatever petty thing it is you're choosing to post. I think you should tell him how all this makes you feel and if he does something about it then maybe it was just him being clueless but if he doesn't then there you go, move on to someone you inspire to be a better man to you.

Honestly, I don't think you are being paranoid. I know out loud it sounds "lame" or whatever to say he "likes" this girls posts and not yours, but I think there is a deeper meaning. Whether its on Facebook or not, your guy seems to seek her out, "stalking" her posts on Facebook, almost ignoring you to help her out, and being a gentleman to her while not doing the same (and more!) for you.

What really worries me the most is he called you her name. My ex did that to me a few times when we started dating, and when I finally confronted him about it he explained it was because when we first got together he wasn't over his ex. Needless to say that relationship did not last.

I think you need to have a conversation with him about it, and ask him honestly if he has feelings for her. And if the answer if yes, then he needs to figure out how deep those feelings are.
People in relationships have "crushes" on other people all the time, it's natural, but it's a problem if he values what he feels for her over what he feels for you.
It could just be a case if him my realising that he is doing it, some people believe that if they are with someone, then it should be clear who they want, regardless of how they act towards others, but if your not okay with it he needs to stop.

I also think you need to think about why you want from this it any relationship, and if he isn't meeting up to that, or won't try, then you should get out of there and wait for someone who just makes you feel special without thinking about it. Being in the wrong kind of relationship isn't better than not being in one. It just messes you up later.

Yes. Time to move on. If you have this nagging feeling about anything.. you're probably right. Maybe she inspires him to be more of a gentleman, and that's what he's looking for. It's nothing that you've done wrong, you are who you are, and you should have someone who wants you just as, not as something else.
Let him go and if they end up getting together and she never mentions something to you, you should let her go too, because that's not what girlfriends do. She should atleast ask you if you're okay with it.
But cut him loose and don't waste anymore time on him, find someone who deserves you. :)
Good Luck!