From the windy city to the rainy city, eh?

Welcome.
I'm a non-native, but have lived here about 12 years, so I can give you a "little of both" perspective. I also don't live in the city, so you should know that. You will get much different answers from folks who've lived here all their lives and/or who live/have lived in the city.
I have lots of friends who bike (bicycle and motorcycle, not sure which you mean) year-round. On average, it snows about once or twice in the winter and the snow doesn't stick around for very long. This year was an exception.
The public transportation is ok. Compared to other cities (SF, for example, where I'm originally from) I personally find it lacking. My wife takes the bus downtown every day and the buses are fairly hit-and-miss schedule-wise. But you can get pretty much anywhere with a transfer or two.
I have never heard of, nor experienced, the Seattle Freeze. People in Seattle are very nice (too nice sometimes - try getting someone to go first at a 4-way stop sign). Maybe this is a little different in the city? I doubt it though.
You might be able to find a decent studio for around that price. Not sure about the location, though. Again, I'm not in the city and haven't rented for years. I'm sure someone else can help more with that.
Here's a link to the Craigslist apt section to get an idea.
I would strongly urge you to move earlier in the summer, if possible, or at least visit. Summer is the only time of year it doesn't suck here weather-wise, and the summer is absolutely beautiful and wonderful to be here. Then it rains for like the next 8 months.

Good luck and feel free to let me know if there's anything else I can do to help you out.

Chris