Does the government have a database of everybody who enters the country?

No! We don't even have any idea how many people enter the United States. It is called an un-secure border!

There is no way to secure the border with either Canada or Mexico and there is in excess of 2000 miles of coast line to the east, 1600 miles on the gulf, 7000 miles of coast line to the west and another 1000 miles in the artic. These are not the actual miles of coast line- but the miles needed to patrol the area, the actual physical miles of coast line total is 88,633 miles. At any give location a person can enter the country undetected and undeterred. When they do- the government has no idea who they are or where they went.

Apparently not for for muslims who enter the country.

Which government? If it does, it's for the government's own use, not for anyone else to see.

They might but they have no way of tracking them or making sure they leave when they are supposed to.

Yes, but it is private.

If they don't, they should.