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This cylinder-like screw is screwed into an opening on the inside of a table. It basically "locks" into the screw that is facing down from the top.
Kind of like this:
I'm trying to get them out, yet I can't find advice because I don't know the name of it.
You took the wind out of my sails with that last paragraph. I was going to suggest using a nut/bolt.
As far as it causing problems when the tabletop is lowered, you can recess the bolt head. Yeah, you could make the cavity in which the bolt will stay on the underside of the tabletop, but maybe you'll drill too far or crack/scratch the tabletop if you chisel it out. It could end badly... which is why you should instead house the bolt head on the top-side of the cross member.
Brown = table top
Black = underlying cross member
The one on the left is what you were talking about. The one on the right is what you want to do.
Also, use a washer under the bolt head, etc.
If you have doubts about the ability to solidly attache the screen with two screws into the wall, I'd consider putting up a 1x2 or 2x2 strip, then attaching the screen to that.
You should be able to use a number of screws to attache the strip to the wall, and if you're not concerned about removing it you could even glue it. Then you can screw the screen to the strip, which should hold it without any problems.
If you need a completely flat mount for the screen, you can build a frame out of the same material.
Best Answer
These are usually a cam lock assembly. The insert needs to be turned about a half rotation to release it from the cam bolt.
Here is a picture of what one of these things look like.