Hi Nshape and welcome to the forum.
1. Yes. Your Drive does not seal the hole. In fact, the Drivewell also serves to drain the cockpit of excess water. As long as the hole doesn't penetrate the inner hull, you're good. It's a good idea to tether your Drive to some permanent part of the boat though in case you accidentally drop it in the water!
2. As long as it doesn't interfere with your seat and you don't put a lot of lateral pressure on it, no problem. Or, if you have space you can just throw it in the cargo well under the bungee cords. Any particular reason you want the Trax cart? It's great for soft sand, but harder to pull the boat in most other situations and requires a bit more maintenance than the standard cart. If you don't specifically need it, I would recommend you use the extra $ for a standard cart, a large rudder and either ST or ST Turbo fins. The boat will work fine with what you get, but the fin and rudder upgrades will improve your cruising range and slow speed handling respectively. If you decide to get a sail later, both will significantly improve the boat's sailing capabilities.
PS -- If you're Nshape, you'll want the Turbofins!