The axes now have a mapping but that would be a separate issue.
Connecting the controller is actually not a bug, this is a hardware feature of the controller to allow you to charge it without connecting to the system as another controller, it also allows you to assign the players in order.
Nevertheless there is nothing your computer can do to tell the controller to connect if you don't press the PS button.
This issue should be closed.
Nothing extracted yet.
Playstation controllers generally work well in big picture mode. There are some opportunities to make the experience better:
Thanks for your great work!