You already said that you can't glue down -- is that really true? I have a floating floor over cement and it's been fine, but I did have it laid by professionals.

The quarter round (the round bit of wood attached to the baseboard) looks like it is well attached (plenty of paint). To get a click-together/floating floor in there you will have to do one of two things: 1. Remove the quarter round (most likely damaging it beyond repair), install your floor and then purchase/paint/install the new quarter round. When you move you will remove the floor (reverse the steps as stated above); or 2. Leave it in place and put the new floor up against it with a tiny amount of expansion gap and ignore the "bad install" because you will remove the floor when you move anyway. If it were me and I HAD TO HAVE new floors (as in my OCD couldn't handle another moment with the vinyl tile) I would ignore the baseboard/quarter round and install the floor up to the edge of the wall as best as I could. I would do this KNOWING this is a temporary floor that I will rip out when a I move and thereby leaving the space exactly as I found it...leaving the quarter round entirely untouched. Good luck. Again, don't do anything until you talk with the landlord.

As G&S suggests this is a common repair. A flooring professional can cut out those planks and drop in (after trimming up the) new pieces with some glue. It is better to cut and glue than it is to 'undo' the floor and reinstall the floor. As for the transition strips, a floating floor with an HDF core will have issues with long runs. Traditionally the longest run allowed for HDF with that type of locking mechanism is 30-35 linear feet. You are allowed to push it through a 'breeze way' so long as the LONGEST run is NOT more than 30' long. If you have not had issues with the floor running through the breeze way prior to this, you probably won't have any issues after the fix either. Do you have the extra planks to go forward with this repair?

You should not have any issues with either locking systems as long as the edges are not damaged. They should not be tapping the edge too hard to the point of crushing and deforming it. The drop lock has it's own issue. Any missing spring will prevent the system from working.