Hi compsci, I did my undergrad at UW, so I've lived in Seattle since 2011. For a studio in the U-District (the neighborhood adjacent to UW), or another close neighborhood, you can expect to pay upwards of 1200/mo. You might consider looking at sharing a place if you can stand it. Housing in Seattle is pretty competitive now, so you'll want to start looking at leases for the fall asap. Currently, I rent a room from a couple and pay 600+utilities in a part of North Seattle that is not considered a "nice" neighborhood, and I was pretty lucky to get that price.
Other neighborhoods to look at would be Ravenna, Fremont, Greenlake, maybe Capitol Hill or Ballard. You can get to UW from any of those places with one bus ride.
Also, don't worry about earthquakes lol, I've lived in Washington my whole life and can remember maybe 2 earthquakes I felt.