How to see what alien types I’ve already captured alive in XCOM

I don't believe there are any stun-capable weapons/soldiers in the demo - which is weird since they make a big deal out of asking you to capture one alive. In the full game soldiers have specific gear that allows them to stun and capture aliens (although from what I read, it ain't easy!)

Your third example is a bug, I believe. I've run into it, it's super annoying (and occasionally devastating).

Unfortunately, I have never seen anything from the UI/options that would help in either situation #1 or #2. About the only thing you can do in these situations is to move to the closest possible cover within the blue, and then if you can't take the shot you want, either go to overwatch or possibly re-position as part of the dash with the goal of moving into a more advantageous position on the following turn. The cover thing is especially vexing, I feel your pain there. The game is forcing you to eyeball it; unfortunately, cover is at high enough level of abstraction such that eyeballing it is not always possible.

Note that Assaults, once leveled, are really good to use in these situations. You can move to wherever you think is best in the blue radius and then if you're unsatisfied with the results you can pop Run n Gun and advance further and then shoot (and you can even do Rapid Fire after doing a run and gun, it's quite a potent combo).