I have no idea which Trainer version you are using or which DOOM .exe version but I'm using DOOM Trainer (PATCH 12.08.2016 + VULKAN) and DOOMx64.exe 184.108.40.2069 and I don't experience any keypress problems that you're describing. I can toggle any of the Trainer features on/off at will using the numpad.
I would answer your question but I don't think you're using the same DOOM Trainer version or the same DOOM game .exe as I am.
I'm using DOOM Trainer (PATCH 12.08.2016 + VULKAN) and DOOMx64.exe 220.127.116.119 (Steam patch).
How would I check which version of the .exe I'm using? Cause I know I'm using the same version of the trainer.
Edit: I'm also using the most recent update of the Steam version. It might just be that I'm running it in 32-bit, rather than 64-bit, but I have no clue how to check that.
[Edited by timmage, 1/2/2017 1:40:08 PM]