1) Most mtb rims will accept 1" to 2.35" 2) The 1.6" has a rating of 45-75 psi, the 1.25" has a rating of 65-100 psi. I don't know if a garage car pump can produce that or not. 3) The tube should be slightly less than or equal to the tire size, so theoretically a 1.5" to 1.9" tube should work for a 1.6" tire.
Most garage compressors can produce about 120psi but may be regulated lower to suit their needs. A bicycle floor pump is a good investment for any cyclist.