I just played it for 10-15 minutes and it all seems working to me. Ammo, No Reload, health and Inf Flashlight all worked anyway. I didn't go far, just enough to try it out. I'm still on a 32 bit Windows (few more days).
Make sure your version matches, turn off virus and firewall and run them As Admin.
Don't forget to choose which version you're going to play at the top. Either DOOM 3 or DOOM 1 & 2. On the lower right of trainer it should say For Version 184.108.40.20656.
I do not know why windows 8 thought the ".tmp" file is a stand alone file to run, so it asked me if I want to go to the marketplace for windows 8 finding program to run the .tmp file if I turn off the anti-virus software.
If I turn on the anti-virus software(short for AVS), the AVS knows that the trainer exe file is running the .tmp file, so the trainer started just fine. BUT, the AVS puts the .tmp file into the sandbox, that is why the trainer is not working.
I just put the .tmp file into trusted file list, and everything is working.
Well, this is my case, hope this can help anyone who have problems running the trainer.