With the weight of the device being taken elsewhere, one needs to consider the thermal expansion of whatever it was attached to, as this may introduce mechanical stress on the casing of the new device that wasn't there on the old one. However, without a General Arrangement drawing, it is difficult to comment further.

The flange used to attach the motor to the driven load, installing a foot mounted one will create missalignement and vibration issues between the driven load and the motor and its mountings.

How about just not using the feet? Maybe they can even be removed.

If the face/flange mount from the old motor fits the new one then no there will not be any problem.

If the motors have the same manufacturer and frame size the end bells are interchangeable. Most modern motors the feet are removeable.

This is a mechanical issue, not an electrical one. If you can fit it and it's done properly it will be OK.