Sounds like a successful shakedown trip. Most of us had a lot more grief than that on our first trip in the new rig. I popped a tire (my fault) and learned that new RVs don't come with lug wrenches. It was a long drive to find one. While I was gone, my sister in-law used the microwave to heat something up and didn't notice that the carousel was still taped down, which burned the motor out on our very first trip. We also had power issues and ended up just leaving and not even staying the night.
To answer your questions:
1a: If nobody used the toilet, then there shouldn't be anything in the black tank.
1b: Give it a shot and see what you think. The hygiene of the gray tank really depends on how lazy everyone was before you did dishes. If they scraped the plates real well, then the gray water shouldn't be too gross. Usually though, gray water has enough chunks of pasta salad, greens, marinara sauce, etc. to build up a funk. I don't know where you plan to dump it or how warm it will be, but you can end up with a nasty swarm of flies if you dump a puddle of food goo on the ground. I've pulled up to camp sites where the previous camper had dumped his gray tank and the site was so disgusting that we had to move along to another camp.
2: The batteries that came with your used rig are almost certainly shot, especially since the little old lady used it so seldom. You'll need a couple of new ones. Go cheap on your first set, because as a new owner you'll probably be pretty rough on them until you get your camping legs under you.
This is really good stuff. It sounds like you're off to a great start. Keep this up until you master the basics, and then you can tackle the nice-to-haves, like lighted camp markers, killer outdoor sound system & theater, solar powered ice maker, etc. The sky's the limit.
