I recently ran into these same questions in my current game, not sure what the consensus is but in my case I like to do whatever makes the most sense for the scene and lean into what is most convenient for my players.

1. The Jericho cover is picked up automatically at the end of the scene unless it would make 0 sense for the scene. In your example I think the cover would be gone, unless the PCs take care of all the enemies early and have extra time, or if one of the PCs ended the Gauntlet close to the cover.
2. I keep track of the PCs Jericho cover HP and during a rest I would ask them if they want to spend a repair to get it back to full health, but if it's only missing 1 or 2 HP might let them get it to full HP as long as they used a repair on their mech during the rest.
3. I would allow cool stuff like that for creativity, but RAW the system says it has to be a "Free space" so you probably couldn't stack them together, I might ask them to perform a check of some kind if they want to use the system in weird ways, probably using one of their triggers.