Writer here. I've studied film a bit and it's been useful, mainly in making me aware that there is a camera, even in written works, and that you should move it through the scene with purpose and care.
I try to write my action scenes like the best movie action scenes. Often they are the best movie action scenes, or at least they start that way. Like, I've literally got them playing in slow-motion in VLC while I write down the blow-by-blow in Notepad.
I've found that written action scenes can't go on for too long, because they're more tiring for the reader. They have to imagine the whole thing, while a moviegoer can just sit back and watch. That's the main difference between novels and films: one is linear and takes effort, while the other throws all this information at you that you can't possibly catch every detail -- and you're not expected to
Funny background scenes in movies? Sure. Funny background scenes in novels, not so much. When it comes to the written word there is no background.
I think if you study one art form, it can't help but improve your understanding of other art forms too. There's a cross-training effect and it happens with every combination you can think of.