Public transportation runs 24 hours a day, 7 days a week. You can get anywhere in the city. In addition, there is Uber, Lyft, and taxi cabs.
Yes, however you have to catch the shuttle from the airport to another location to pick-up the cars.
Please stay away from the Circus Circus or the Strat. They are high crime hotels that should be avoided. Typically, they are the cheapest, but remember you get what you pay for.