The cost of living in Seattle is definitely higher than average, but I think that $2250 should be enough to comfortably live on, if you are planning to share an apartment with someone else. When I lived near UW, I was paying $1200 a month for a one-bedroom, but I didn't share with anyone. There are definitely 2-bedrooms available for around $1800, especially north of campus, so I would estimate paying around $900 a month.
I might be biased because I've lived here for so long, but I really wouldn't worry too much about the earthquake danger. I would avoid living in any old brick buildings, but overall Seattle is built to withstand most earthquakes. I don't want to say that there is no risk of damage from earthquakes, because that's definitely not true, but I feel perfectly safe living here and having an earthquake preparedness kit in my car. In the one earthquake I have been in here (6.8 magnitude), one person died, and it was from a heart attack.
Believe me, I completely know how you are feeling! It is a difficult decision to make, and it's good that you are considering these things. I might be moving to Miami to start grad school in the fall, and I'm a little worried not knowing what to expect about hurricanes!
Let me know if you have any other questions and I'll do my best to answer them. Congrats on your acceptances!