Will I then be able to send right/left keyboard strokes that will affect whichever application that is active at that time?
No. You will be able to read from the Bluetooth Mate as if it were a serial port. You would have to write a program for the PC that would translate whatever serial messages are received from that serial port into keystrokes.
The Arduino Leonardo and other ATmega32u4-based Arduinos can emulate a keyboard if directly attached to the PC via USB.