Very cool! I have yet to get used to the sensitivity of the mouse and Git Gud, but this is really fun.
Edit: Ok, now that I finished the game on "Hey, not too rough", as I never played Doom before, let me tell ya this: the game is a freaking masterpiece. From the levels, which are easy to navigate, to the enemies, fantastic sprites, to the music, bombastic at the start, sorrowful at the end, the secrets, never too obvious but still hinted at, the Cyberdemon fight was also really cool... If I didn't know, I'd say this was an official Doom title! Despite only having 6 levels, the game feels full and worthy of the title. Fantastic job!
Edit: the one new thing that comes to mind might be the low count of songs, but I don't know how are the limitations of PICO-8 and the amount of space you've had left after programming the engine and maps and such. The placement of the songs is great though, but I still think hearing "At Doom's Gate" for 3 consecutive levels might be a bit too much.
One thing, though. I didn't get the respective medal for beating the game... Only a secret medal, the cartographer. What did I even do to get that one?
Answer: I do? Funny, I can't see them... Are you sure about that? Anyway, how the heck did I even get the Cartographer Medal? Getting every item? I did find every secret in all levels, so there's that...
Finally did it on "hurt me plenty" mode! And all the secrets, in one go, all demons slayed and only died once to the Cyberdemon.
Double Edit: Ultra Violence. Beaten. No deaths.