If you have a hard time visualizing, just think of it this way: The mitre cut gives you the "edge" that will sit perfectly on the straight piece...but to get that edge to rest on the straight piece you have to remove the excess wood behind that edge (via coping). So just make sure to leave that outside edge alone as you cut- mark it with a pencil first. But if you're going to paint the base molding, why not just fill the inside gaps with paintable caulk? If you go this route, start the mooldign on outside corners and make sure they are tight with each other (like going around a support pillar or outside closet edge, etc.) (I brad nail outside corners together so they stay tight.) Then fill any inside corner gaps with latex caulk and immediately smooth with a wet finger. (BTW, you also fill any gaps between the molding and the wall with caulk.) Keep a bucket of water and a wet sponge nearby, and keep wiping your fingers as you go. If your'e not painting, then definitely it's better to cope.

If you haven't already paid for the installation, withhold payment. Call the remediation company, and put it in writing if you can also (memorialize the call in an email). They might direct you to someone at the flooring company, or, considering they have been contacting the flooring company on your behalf until now, they may take care of it themselves. If they are not helpful, call the insurance company. Peeing in front of you and not washing up--during a pandemic, no less--is absolutely grounds to never let him in the house again. Not to mention his unprofessional and abusive conduct. Don't be shy about putting your foot down, and don't compromise just because you've already had to wait a long time for this to be done.

@Ana Gutierrez: It’s least worth your time to research engineered wood. It’s not as expensive as people think and many handy homeowners are able to install it. It also goes over some types of existing floors. It looks the richest, especially if a medium brown is your preference. Laminate floors are the least durable and the worst in a wet climate. Any flooding, even a little, will ruin it. Luxury vinyl tile gets great reviews. The better products look more like wood than laminates. All of these can be bought in planks that are not hard to install. Tile, as was well pointed out, is ridiculously expensive and hardest on your back. Most vinyl will survive the dogs pretty well. I know a family with two Rhodesian Ridgebacks who installed luxury vinyl in the bedroom and bathroom of their twins’ suite. The mother wishes she had a reason to put it around the whole house. Do your research. Ask questions at flooring places so someone knowledgeable can answer about durability and dog-nail scratching. You can shop for bargains after you get all the information together. Don’t go to big box stores yet.

That's the floating part. It is supposed to move (expand and contract).