Those are 5V relays. Have a fire extinguisher handy. You will smoke them with 12V.
Try removing the jumper and powering JC-Vcc with the arduino +5V.
With the jumper installed you are powering both the relay and the optoisolator (which isn't isolating with jumper installed) directly from the microcontroller pin. When you remove the jumper the optoisolator is powered by the output pin and the relay from the regulator.
If it works, good. If it still shuts down, then we know the regulator is overloaded.
Yes the 7805 will work, just use it to connect to the 12V supply you already have.
The suggestion for the transistor/MOSFET was to make a relay driver. But your board already has one, but with the jumper installed it is bypassed.