Congrats on your fish!
1. What kind of food do you have now? You should feed once per day, enough to make his tummy look a little rounded without looking really big. And you should feed a good quality pellet without a lot of fillers - something like Omega One or New Life Spectrum pellets.
2. That can be normal. Bettas rest sometimes, and mine rest at the bottom or on a leaf or wherever they feel like. He should be swimming around often though and not resting all day long. Also, unless you live in a tropical country (and even sometimes still then), you will need to get a heater for him. Bettas need temperatures of at least 78 degrees.
3. The light is really up to you. If you don't have live plants in his tank, and he gets enough ambient light to have a normal day/night cycle, you don't need a tank light. Tank lights are really mostly either to show off his pretty colors or to grow plants.
4. Hyper is pretty normal as long as he's not wearing himself out or bumping/rubbing/scratching against things (this is called "flashing" and can be a sign of parasites).
5. Yes, my bettas swim all over the tank, including the bottom, top, and middle.
6. Gravel is really just for looks. If you're happy with what you have, then it's fine. It can be easier to clean a tank without a lot of gravel.
For some added advice: I think that the little cube you have is only 1/2 a gallon, correct? You should really consider upgrading to something a little larger - at least 2 1/2 gallons. Not only is it difficult to keep a smaller tank clean, it is also difficult to keep it heated consistently. Depending on your budget, we can make suggestions that may cost you as little as about $30 to give your little buddy a nice upgrade to make him more comfortable.
While he's in the little cube, you'll need to be changing 100% of the water every day and rinse the gravel as well.
