Hi! I've been trying to identify what is causing vibrations coming from roughly my rear-derailleur area when riding over bumps.

I'm a newbie and don't have another bike to compare, so I'll ask here. Is the movement here right or too loose? Should it even move? Even the chain moving up and down moves the entire derailleur around the screw holding it to hanger. I seem to not be able to rotate the screw any further without messing up the head. Am I missing something here?
Someone said that my chain path may be wrong, but I can't see anything wrong?

Video of what I'm talking about: streamable.com/mqy2jj

There is a pin located just above the upper pull chain jockey pulley that locks the pulley cage forward to make it easier to take the wheel out. That pin needs to be released - probably by pushing the lower pulley forward and maybe pushing the pin with your finger.

Hi, thank you so much for the answer!

I guess I wanted to show this better and thought that pushing pin in will show the problem better, but it just makes everything way more loose than it is. I've made another recording, at 1/8x speed to show off how the entire rear-wheel with derailleur behave when dropped a bit. Is there too much movement from somewhere? Again, I don't really have anything to compare it to.

New video: streamable.com/aavkj7

To me, it looks like the chain is too long, but this setup worked earlier perfectly fine. Now, I feel every bump, even ones as dense as the tactile-paving markings for blind people.

Looks about normal to have enough chain to go over the largest cog.

Get a pic of that.

Also, is the clutch engaged?

Looks fine to me - I also think that your chain might be a bit too long, but I'd need to see the chain on the largest sprocket to make that call

For a quick in situ chain length check-
Biggest Ring & Cog.
Have at least 1" of slack.