Remember last open directory
Since there is Home key to go to current dir and Shift+HOME to go to root
I am not sure what you mean by this. file-browser already remembers what the current directory is when you open and close it. Are you saying you want to be able to jump from some random folder to the root, then press a button and jump back to that random folder, wherever it may be?
When I close and reopen mpv, pressing option or ctrl+o opens the root folder. Can it remember the last directory I opened, even if it's not the root directory?
Theoretically this feature could be implemented, but currently it's not supported. I'll consider implementing this if I find the time, but in the meantime I will leave this issue open.