I find it confusing to try to identify who "I" am. I don't know the difference between "me" and a part. The way I currently perceive myself is as a bunch of parts, some of whom have started to think the same way as another. To me, this seems like a step toward integration.

You are not speaking gibberish; this stuff is confusing. I have had experiences similar to yours where suddenly stuff would be coming out of my mouth and I would be thinking "who is saying all of that?" For me, parts being willing to be silent while another was talking was one of the first signs of cooperation among them. While I agree that it was necessary for me to trust the T before I could do the parts thing, I also had to develop some trust amongst my parts.

At this point, usually my parts know when they are "out", and generally the rest of me (whoever that is!) knows which part is out. Sometimes I react in the moment, and then stop and check on who was in charge. Ts and H spot changes sometimes before I do.
So I guess my answer to your first question is that yes, parts can know when they are out. But they don't always.

One thing that I have found helpful is to invite a part to journal. Literally ask myself if the part that was at therapy that day if she would like to say anything, and then just write or type it without analyzing or even really thinking about what is going on paper. Then I ask if anyone else wants to speak. I usually take what I write in to T1; he likes to see what is going on and can make helpful observations. But I don't think that is necessary.