In the States it is known as CTAF (Common Traffic Advisory Frequency).
1. No. I've never worked in a tower or approach that didn't operate 24/7/365. It does happen at airports near mine where the towers close at night.
2. In the States the number of commercial aircraft that will fly into an airport without an operating control tower are less than a handful. Most airlines cite insurance and safety reasons. Occassionally, a commercial flight will run late and the tower will stay open a little later (paying some overtime) or just close but the aircraft will generally stay with approach or center until on the ground.
3. Of course safety is being compromised. There have been numerous mid-airs in the States around airports without operating control towers. I can list 4 in just the last 6-8 months in SoCal. None involved commercial aircraft.
4. I believe you have to be a U.S. citizen to work ATC in the States.