I don't know if I can answer ALL your questions, but I'll tell you what I know... my cousin is a chef/caterer, and he went to the California Culinary Academy in San Francisco (Le Cordon Bleu). It's known to be a very good school, and my cousin's had a successful career! As for the neighborhoods, there are MANY Asian communities in the Bay Area. Highest populations are probably in South San Francisco, Daly City (mostly Filipino), San Bruno, and parts of San Francisco - Sunset District, Chinatown & Japantown, Inner Richmond, etc. In the East Bay, I'd say that Alameda and Berkeley probably have the largest communities. I'm not sure specifically about Vietnamese, but they're likely in most of the same neighborhoods - just in smaller numbers. According to City-Data, the ethnic breakdown of San Francisco itself is 19.6% Chinese, 5.2% Filipino, 1.5% "Other Asian", 1.5% Japanese, 1.4% Vietnamese, and 1.0% Korean. Anyway, my point is that you won't have trouble finding an appropriate location! And Asian restaurants in general are very popular here, among people of all ethnicities... my people (I'm Jewish) in particular are frequent patrons.
If you want to be near the center of action, the San Francisco neighborhoods of SoMa (South of Market), Inner Richmond, and Japantown area might be good... other districts are slightly closer to downtown, but very expensive. Though finding a place anywhere in SF for $500-1200 would be tough!! With a roommate it could be possible, or you might find a studio apt. for around $1000. The East Bay is usually slightly less expensive, whereas the Peninsula varies from less to much more... North Bay is too long of a commute, so I wouldn't suggest that.
There's also a Culinary Institute of America in Napa Valley, which is a great place to live... plus I've seen the campus, and it's gorgeous! You can see their website here -
http://www.ciachef.edu/visitors/gs/. But I'd guess that a chef would have a lot of competition, due to the wineries (which usually have amazing restaurants) and weddings/events around there. On the positive side, it is slightly more affordable than the Bay Area! Anyway, I hope this all helps a little...
