But, if I have something connected that uses a lot of power, using an external suply, and this PSU breaks, the arduino will be using the usb suply, and I could break my mother board... Am I right?
The potential is there. The Arduino boards do have an on-board polyfuse connected to the USB supply to help in this type of situation.
Maybe I should leave the arduino being suplied through the usb and suply the other board (the one with the shift registers) through the external PSU, and then just connect the grounds on the usb and on the external psu?
Personally if anything I have connected needs to draw more than 1A, I don't use the Arduino board as a supply. In that case, it makes more sense to supply the other components directly. And yes, you'll need to connect the grounds (aka "common".)