FWIW, it's usually Windows Installer, Windows Explorer, or some other system task. If you open a command prompt and ready the del command, you can kill Windows Explorer and then run your delete.
second, I'd get powertoys for windows, it has a feature that shows you what processes are holding onto a particular file/directory so you can kill the process and get the file dealt with.
I think I've had a similair problem before! Turning off the "preview pane" might help if you have that option enabled. I've had trouble with renaming files that were also open in the preview pane in the past, but turning it off fixed the issue
Looks like a restart fixed it, but I think it refused to delete since you had the folder open. Your task manager showed 2 instances of windows explorer. I bet it was minimized and got overlooked
Almost seems like you might have some sort of virus or malware.
There's a magic registry key that can force delete folders and files before Windows has even finished booting. It's intended to be used during Windows updates, but when used creatively, can perform some brute force fixes.
This link isn't exactly relevant, but I'm not home right now to dig that deep into the registry to offer that sort of advice at the moment.