Have you called AS or NW to see if this is a legal connection at GEG?
I wouldn't count on making your connection in 30 minutes. I've done the C - Main terminal switch a few times and sometimes breeze through in no time, other times there's a 30 minute security line at the Main terminal. I assume you're on a single itinerary, making connections at GEG, not two separate tickets. If so, you'll already be checked in for the NW flight, I'm not sure they'll give away your seat at 20 minutes, they'll know about the tight connection.

Good luck!