I think Doom doesn't fire from the tip of the weapon, I think they shoot from the center of the screen, so you don't need ADS because you are "always ADS"
Your guns are not aiming at the same spot the cursor is. You need to dinamicaly find the intersection of the center of screen line and the enviroment is to point the gun that way.
theoretically (given that your guns are not parallel to the camera direction) the bullet and the cursor should be pretty similar in very long distances, but as you added spread to the guns shooting from far away makes it impossible to hit. And them shooting from close up there is this noticible inaccuracy to the guns
Yeah making 3D games is a beast in itself, we worked for some months in a multiplayer FPS, but ended scrapping the project as it was too much