I'm not one of the people that actually does any research on the tesseract, or affiliated with SHIELD, so take this with a grain of salt, but this is what I know.
People come here through the tesseract, the way you just did. They can come through from any point in their timeline, so far as I know. They can last here as little as a few days or up to a year-- people started showing up last May, by this world's timeline. Whenever they disappear, it's just poof, gone. But they come back, sometimes. They never remember being here before, though, and I don't know why that is. I also don't know why some people leave so quickly and others stay for a long time or why some come back and some don't. I just know that all of the above is possible. Leaving, staying, returning. And while we're here, we all just kind of try to find a way to get by in this universe.
I'll try to answer other questions if you have them, but like I said, not exactly an authority on the subject. I've been here six months and I'm observant, is all.