It is acceptable to put the subwoofer in a cabinet, provided that there is an acoustically transparent opening for the sound pressure waves to radiate outward.
Yes, you should put something soft underneath the subwoofer so that the spiked feet do not rest on the hard surface. Probably the most convenient way to orient the subwoofer is so that the port is facing outward (so that one can see the amplifier panel when opening the cabinet door). Try to give at least 1-3" of extra space on each side of the subwoofer. Perhaps you can use some foam pieces or pads on the sides and top of the sub to help reduce some resonance from the built-in cabinet unit.
Pete - Hsu Research