(I can still only post two links per post, apologies in advance.)
As @ebassi mentioned, connecting to the changed signal is the easiest way to be notified of settings changes.
In the Python Console plugin, you can see the plugin class connects to self._settings’s changed signal, then in the signal handler (on_settings_changed(), further down the same file) it reconfigures based on the updated settings.
You can also connect to changed::x that is only triggered when the x key is changed. I suggest reading the documentation for more details.