1. is the setup itself sufficient, and how do I check?
Just wait. Start with the lights on for 6 hours a day and see if there is algae growing or if the plants are dying off. Adjust light based on what you see.
2. is it worth keeping the plants or at least some of it or do I start over?
Most of those plants look like crypts, and EXCELLENT beginner and low light plant. Keep then, or, give them to me
3. do I need to treat the water with anything in particular after moving? I took about 4/5th of the water with me.
Taking water with you is a misnomer made by pet stores and salt water tanks. You'll need to treat it with dechlorinator if you are using tap water to fill the tank. Check Amazon for "Prime", it's the best.
4. do I clean the gravel, and if yes, how?
You can, but it isn't necessary in my opinion. A gravel vacuum on a hose/siphon works. But the plants enjoy the nutrient rich substrate. I usually pick out the dead plant matter and don't worry about the rest.