You can tape cheap shower curtains around the back and sides. If you can’t find cardboard big enough for one door you can have two doors. Or each shelf have its own door.

Can you call the movers and ask what they did to keep the door open? Maybe one of the guys put the pin or screw in his pocket and forgot. He may bring it back. If nothing is missing they may assume you know how your door works. And may (should) be willing to come back.

If you mean you want a place for the dog to do it’s “business” then pea gravel is an option. Definitely put a weed proof thick cloth under it. My warning is this. If it’s windy and there are trees and shrubs nearby that shed seeds and leaves you have to clean that up quickly. Otherwise it will fill up with baby trees, grass, and weeds.

You might be going too fast. If you’re pushing the saw faster than the blade can nibble the wood it will bend. Go slow and keep your eyes on the spot where blade cuts the pencil line. It that wood is very hard this will take a long time.

So the top will be higher than the two sides. Even after the remodel? What about the top several inches of the dishwasher? Box it in? For the remodel, seems like the dishwasher should be installed on the floor and route the heater under the toe kick of a cabinet.