It's not a stupid question (though you really need to give more information next time!). The LM393 indeed should be able to handle up to 36V, and I don't really see what the problem is.
At 12V, the 1k resistors will dissipate about 100mW - a bit high, but shouldn't be enough to pop them. That was my first thought, maybe they're burning open circuit. But no.
Are you sure it's actually an LM393, and they didn't use another similar chip on your board that might not be rated for 36V? The LMV393, for example, is only rated to 5V.
Are you sure you didn't do anything else to mess it up? You'll piss it off if you connect a voltage directly to the output, for instance.
As for the MOSFET - 12V at 1A is hardly anything for a modern FET, just about any TO-220 MOSFET will do. Just make sure the gate threshold is low enough for it to switch on at 5V (keyword: "logic level"), and obviously, check all the ratings in the datasheet.