Yes, the 1600 can be cross-flashed into a 1620 (the 1610 is a different drive and can NOT be flashed with any other firmware).
The 1625 is a lightscribe drive, so cross-flashing to that firmware won’t get you anything, because you need actual lightscribe hardware.
The 1620 Pro is the same as the 1620. That’s just a marketing name that has no meaning.
The 162I and 162U are external versions of the 1620 (actually, I’ve never heard of the 162U, so I’d have to check to see what that is specifically).