Can an Arduino (Leonardo) be recognized via Bluetooth as a mouse/keyboard?
If I buy something like the HC-05 how would the chip report itself?


As an SPP. I.e., a serial port.


  How is the bluetooth stack handled with regards to identification and authentication?


By the firmware on the HC-05


  Does it somehow copy the usb mouse/keyboard setup?


No.

You need a USB device that can emulate a keyboard. That's not the HC-05. The host microcontroller has absolutely nothing to do with what capabilities the bluetooth interface has.