Sounds like you have a few things going on here. Scat mats - no. I don't use penny cans either. For one of our difficult cats, I used a small squirt gun. I only needed to squirt her once or twice, and she just needed reminded with the sound. (She wasn't on the counters. She was waking me up at 4 a.m. when the birds started chirping) I think some folks use a squirt bottle for dogs, though I don't, and it would mean you need to be right there when Truman jumps up. Same for the cats. Personally, I would take the dirty dishes out of the equation. Problem solved except for the kitties. I like the clorox wipes a lot. I think they kill about everything. We like baby gates, but we've only needed them for a short time.
He's a puppy and going to be one for a long time. When I trained our puppies and helped my mom train my brothers' puppies while I was still at home, we never let them out of our sight and we had regular schedules for everything. I wouldn't use belly bands on a puppy (Puppy Advice thread). I learned early that the more time I put in when they are young, the better dogs I have when they are grown.
Jake came to me at 1 1/2 years old. He was pretty together when he got here, but he's still a pup and forgets proper behavior sometimes, like jumping on me when I come in the door. He is now 4 and still forgets sometimes. We still work on it.
You've gotten good advice from the others, especially the puppy raisers. Good luck with Truman.