I am looking to install a 3/8" engineered hardwood over my existing 19/32" OSB subfloor. The existing flooring to be removed is a combination carpet/hardwood/tile.

The manufacturer recommends a glue down installation (although staple is acceptable). Am I okay to glue this directly to my existing subfloor? If I need to remove it in the future, will I then also have to replace the subfloor? Seems so based on the sticky, but just wanted to confirm.

Alternatively, would it be okay to install a 1/4" luan (or alternative) over the existing subfloor and glue to that? This would also increase my floor thickness and solve my issue with too high cut (from existing flooring being taller) door casings. If yes, should I put anything in between the two layers? If not, an other ideas?

You are correct in assuming that anything glued to the subfloor is going to take the subfloor with it when you try to remove it. Adding luan and then gluing to that would be better for future removal if the extra height isn't an issue.

Nothing needs to be added between the luan and the existing subfloor, though you will want to make sure that the existing floor is smooth at all joints to avoid any noises after installation. Make sure that the existing subfloor is secured to the joists well, you won't be able to fix squeaks easily once the new floor is down.

You don't want to glue directly to the subfloor since it runs under the walls this can make removal very tedious and it's tedious enough. Plus I hear it weakens wall structures if removed-Not positive about that.
Gluing to luan is not recommended either, get some 3/8 or 1/2 inch ply.