Photos Library.photoslibrary constantly re-appears in Trash

Where did the Library come from? Where do you expect it to be? What is the size of the trashed Library?

You could try starting in Safe Mode to see if that fixes it. This bypasses certain potentially disruptive processes. Safe Mode is different for different computers, so see this: 

           Start up your Mac in safe mode - Apple Support 

By the way, when you add screenshots (which are very useful) you can make them smaller by grabbing the handles at the corners.

The normal location for the Photos Library package is in the Pictures folder. You should try dragging it there.

I asked the size, because I wanted to know if this was an empty library package or if it had pictures in it. Have you put pictures in it?

What happened when you restarted in Safe Mode?

---- Try using the option key with the Finder menu "Empty Trash"

When this is in the trash, if you control click on it, you would normally get a pop-up menu like this:

With "Put Back" as a choice. Normally, choosing that will put the item back where it came from. Of course, that may not work this time. Do you see it, and does it work?

I think that may be what you showed in your first screenshot, and "Put Back" wasn't there, is that right?