What requirements are needed for a soldier to use the Gollop Chamber

Adjacency is horizontal and vertical. Each "pair" of facilities counts as one adjacency bonus. Generally you can count the plusses to determine the specific level of benefits you are receiving.

The types and adjacency bonuses are as follows:

1. Laboratory: 10% research speed bonus for each adjacency bonus.
2. Workshop: 7% refund on resources used to build items (that's gear, facilities, and vehiciles) for each adjacency bonus.
3. Satellite Uplink/nexus: +1 capacity for each adjacency bonus.
4. Power, Elerium, Thermal Generators: +2 power for each adjacency bonus.

Note that if you go into the Build Facilities screen, you can see this in the description text (of the currently highlighted facility), but sometimes you have to sit there and wait for it to start scrolling.

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).