Does supinated-grip press carry less risk of shoulder impingement than the press? The elbows seem to travel more in front of the body when compared to the press - and so the shoulders seem to be less internally rotated than during the regular press. Does the underhand-grip variation of the press carry less of the risk of tearing the rotator cuff relative to the press?
No. Neither variation carries any risk of shoulder impingement. Shoulder impingement is associated with end-range rotation, and is classified into internal or external shoulder impingement, depending on to which extreme the humerus is rotated at the time. Both variations of the overhead press leave the humerus in the middle of its range of rotation.
The elbows seem to travel more in front of the body when compared to the press
That's because the bar requires more forearm supination than you would otherwise be capable of, and the only way you can complete the movement is by pointing your elbows forward more to compensate, so that your hands don't twist off the barbell.
You can test this by trying to perform the movement without an barbell, just empty hands grasping an imaginary bar. Try it with your elbows forward, attempting to keep your palms aligned so that they're both facing directly backwards. Now try the same with your elbows out to the side. You'll find that as you raise your arms, your palms rotate to face each other. They couldn't do that if they were actually grasping a bar, so you wouldn't be able to complete that movement.
and so the shoulders seem to be less internally rotated than during the regular press.
The shoulders are not at all internally rotated during the regular press, so I suspect you're mistaking what internal rotation is. If you hold your arms in the position they'd be in in the middle of a press, then while trying to keep your upper arms in the same position, move your hands towards each other until they touch. That's internal rotation. External rotation would be again holding the same upper arm position, but trying to move your hands as far away from each other as you can.
Upright rows are an example of internal rotation in a similar upper arm position.
Does the underhand-grip variation of the press carry less of the risk of tearing the rotator cuff relative to the press?
No. It just makes it more awkward and forces your elbows to a forward position, which if desired is something you could always voluntarily do in the conventional press.