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There's a really jarring delay between the enemy spotting the player and firing at the player - like they fire at your previous location and not where you are now.  How can I fix this?

Also, you should replace the Doom assets with something like FreeDoom.  That way you won't get hit by any copyright issues.   

P.S. This is a great starting point for making a doom clone in Godot.  I love it!

Is there any way to add gravity or jumping?

If you're talking about directional/rotational sprites or whatever they're called, there's an example and also a youtube video. 

There's a collision example.  Also, if you're using sprites, you do collision the same was as in 2D Gdev:  Separate Objects.

So if I just want to make a doom clone (only look left and right, not up and down), what do I do then? I have the extension. 

EDIT: Nevermind, it's in the "Sprite Projection" example! Cool!

Go into "Edit Object" on the sprite, click advanced options, and uncheck the "Don't play the Animation etc. etc" box

I'd like to know too!

Janky, but fun.  I'm now totally convinced that Godot isn't the right engine for 3D, but who cares?  It's another actual FPS game and not a modern pieces of crap where you move like an arthritic sloth and have to use nauseating, disorienting "Aim Down Sights" in order to hit anything.  It's fast, responsive, and the theme and concept is weird as hell.  I love it!