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Caliber: 9mm LugerThreads per Inch: 1/2x28Additional Features: Standard Size
It is personal preference, both models will work as the PAK9 is 1/2x28 threaded. Since it is a fixed barrel, the larger model MPN: LWD-COMP9M would likely be the better option as it will help mitigate more recoil.
No, as these are designed for pistols, they use set screws rather than shims or crush washers.
The Lone Wolf Arms Compensator (MPN:LWD-COMP409) fits LWD 9mm conversion barrels threaded 1/2 x 28, but will not fit any standard LWD 9mm barrels threaded 1/2 x 28.
There is not a Lone Wolf Arms Compensator LWD model for 10mm. However, Lone Wolf does make an Alpha Wolf Compensator specifically for 10mm. They designed it specifically for 10mm in order to provide reliable functionality. Since they have two different compensators for 40S&W and 10mm I recommend using a 10mm specific compensator for a 10mm handgun.
As long as you have a threaded barrel you can use the .40cal comp. The Lone Wolf Arms Compensator offers a 1/2x28 and 9/16x24 thread option.
None of the Lone Wolf Arms Compensator variants are compatible with 45 ACP. There are compensators on the market that are dedicated to 45 ACP, you can find some of them using the following link:
https://www.opticsplanet.com/muzzle-devices.html?_iv_parent-caliber=dot-45-acp
I would recommend using what is needed so it is timed properly.
The 40S&W option would be the correct one, just make sure your thread pitch matches the compensator.