Arduino Uno, HC-06 Bluetooth & RGB LED Strip on the same 9V Power source
Yes. It will. Keep in mind that the 3.3V regulator is limited, and the bluetooth module may take too much current. Most bluetooth modules that work at 3.3V tend to come with their own 5V to 3.3V regulator on board.
Yes, you connect the grounds together. That is expected and required.
It is common Anode or positive pin. While electrical theory is that electrons flow negative to positive, conventional electrical engineering shows positive to negative. The led here has three individual cathode and are low side switched.
That said, 9V tends to be too low for common 12V led strips to turn on at all in my experience, and a 9V battery will not last long at all in this setup. They just arnt meant for high current draws.