If an object's gravitational potential energy changes, does it's velocity and momentum change as well? How?

Maybe, maybe not. Depends on how it is done.





No. A ball is rolling across a horizontal table. The table is lifted (or lowered) without vibration and keeping it perfectly horizontal. Velocity and momentum do not change, though the ball is higher (or lower) and has more (less) gravational PE)





Yes. A ball is rolling across a horizontal table. The table is lifted on one end so the ball rolls up hill (or lowered so the ball rolls down hill) both velocity and momentum change. The ball higher and has more (or less) Gravational PE