Can Obsidian monitor and display recently opened directories.
I hope to see the history of recent directory searches at any time.
Then access these directories again.
And open it with a third-party resource manager, such as TC, OC, Q-dir
Not, as far as I am aware. In fact, directories are only a secondary aspect of Obsidian, only exposed to the user through a core plugin. It is perfectly possible to use Obsidian without caring for file storage and folders. So no, standard Obsidian does not track what directories you opened. The Quick Switcher maintains a list of recently used notes, but that is as far as it goes. There may be Community Plugins, though.