I'm not 100% sure on any of this, but from what I've read here I'll do my best to help.
1. Water, you could use all of it and add fresh mixed saltwater to it.
2. Sand and rock hold your main bacteria for keeping a healthy marine aquarium, the more you save the less time your cycle will take. thoroughly cleaning your sand would remove all beneficial bacteria, thus making a longer cycle time. Also, since it is a bigger tank, there will be a cycling. The bacteria in your sand and rock is only sufficient for the 10ga. So you will need more sand as well as rock to provide more ground for the bacteria to build up on.
3. For the rock, same as above. There will be a new cycle but it shouldn't take too long.
4. definitely add more rock, bigger tank means more rock for filtration.
5. Best to wait at least 6 months before adding coral. The more time your new tank has to flourish, the better success your coral will have.
6. I have mine to turn the Actinic lights on 30 minutes before the daylight light turns on. And reverse the timing at night. IE: Daylight turns off 30 minutes before the actinic. Then when both are off my moonlights turn on.
7. This I'm not too sure on, I have ZERO experience with nano cubes. But from what I hear your choices are severely limited. Possibly some small shrooms
and the like. Anemones....Sebae, BTA, LTA, etc....I think.
8. Hope this helps, good luck, and please let us know how it goes!