Without it being obnoxiously big, I don't think a piece of plywood is going to help save the vinyl flooring. I'd just accept that you're going to leave an imprint behind and be prepared to lose part of the deposit the day you move out.
Moving the fish, I'd treat it like shipping them through the mail or taking them to an all-day sale somewhere. Buy some legit fish bags (
www.jehmco.com), fast them 24-36 hours in advanced, and double bag them all. If you don't have a styro box for transport, you might try checking with your local fish store that receives fish shipments. If they just throw out the boxes, you might get some for free. Paying a few dollars might improve their interest in helping if you're not a regular customer. I'd check in advance of actually needing them though because they might not receive shipments all the time.
The water doesn't matter. Waste of time and a hassle IMO. You need to transfer the beneficial bacteria, which isn't free floating in the tank, but attached to surfaces such as the biomedia in your filters. Bag that up just like the fish and to keep as much alive as possible during transport. If you have more than 1 filter, do this for them all.
Setting up the tank, using all new water is fine so long as the water has been treated if necessary and is of sufficient temperature. The only concern IMO is whether the pH isn't insanely different, but tank raised fish are better at adapting to a wider range of pH than the WC ones.
