ATX PSU's have one -12V cable, and a few 12V yellow ones. Could I power up a 24V device by connecting 12V to + on the device and -12V on -?
It depends on the device. You get 24V but the -12V is usually low power. So you could get a reference voltage. You could use it for powering very low power devices. Check the current rating of the -12V line on your PSU.
If you want to use your ATX PSU to get 24V, you should get a DC-DC boost converter from 12V to 24V. They are cheap and powerful.
Yes, but be aware that you are essentially redefining ground for your 24V supply to be 12 volts lower than the ground for the other power supply outputs (5V, 3.3V). This may also be true regarding the neutral and protective ground of the AC input. You will need to be extremely careful if you make an electrical connection between your 24V device and any other electrical device or to earth ground. If you just want to run a small 24V fan you would probably be OK, for example.