Finder closes after tasks completed

The Finder should always be running, so I don't quite understand how it "closes". Does that mean all open Finder windows close, or...?

How are you reopening the Finder? If the Finder app quits, it should re-launch itself. If it does not, the processes required to reopen it would be unavailable rendering the Mac essentially unusable.

Please determine if the same problems occur in "Safe Mode": Start up your Mac in safe mode - Apple Support. Use it for a while in that mode to determine if the same problem occurs. Then, restart your Mac normally (not "Safe Mode"). Reply with your observations.

Are you looking in the Bin when you empty it?

If so, the bin will automatically close when it's emptied - there's nothing to see here, after all.

That doesn't mean the Finder itself is closing - it should still be running and any other windows you have open should remain. Are you saying that's not the case?