Ah, that was quick
Hmm, I'm sitting with a potentiometer set to 170 K (more or less) and a 1000uF cap, gives me the time I need.
Yes, but with about 35uA average through that pot, you are going to be subject to leakages and bias currents (the 555 will take some current at the threshold pin, and the capacitor will leak as well). The problem is that these currents aren't constant - they will drift with time and temperature. You might have the time you need
now, but in a week it might not be right anymore.
Then again, it could be. That's the problem - using a 555 this slowly is a bit unpredictable. If you use all the right parts and design carefully, it could work, but you need to be careful.
Sorry, I'd try to work out your timing logic but I don't have time right now. Hopefully someone else can help you with that. (And you really ought not to use relays for logic.... 1945 called, it wants its efficiency levels back...)
Don't drive a relay directly from the 555 like that. At least, put a diode backwards across the relay (pointing up to the 12V supply) to absorb back EMF when the coil shuts off, or else your 555 won't last long. And make sure the relay's coil current is within the maximum output current (TI's NE555 allows 200mA, so you should be fine with that.)
I still don't understand why the PC can't shut itself down in software.