First, welcome to Boston!
Sounds like you'd fit in well in JP. I don't know much about lower Allston but I currently live in JP and it seems to fit your criteria well enough. I live in a very large one-bedroom at the top of a quiet hill and I'm within a five minute walk of two pubs and one of the only food co-ops in Boston (the other is in Cambridge). I also only pay $550 a month in rent. I'm very aware though that I am extremely lucky with my living situation because rent around here is usually twice that. However, if you look hard enough you find little gems. The other great thing about living in JP is transportation. The bus runs pretty regularly and brings you through Huntington Ave (Northeastern, MassART, Wentworth, Harvard Public Health, Longwood area hospitals, etc.) and into Copley Square. For a fast trip to downtown Boston you can hop on the Orange line. My favorite, though, is the Southwest Corridor Parkway. It's a flat green space with bike path that stretches from JP to Copley Square, about 5 miles passing through the Huntington Ave destinations I just listed. It's a great option for the warmer months and it gets me to my office near the Garden and Common in about 20 minutes.
The JP crowd is pretty diverse. People characterize it as a lesbian enclave, and it may well be. It's also very family oriented but youthful. There is a fair mix of Haitian, Dominican, African American and Puerto Rican families, mostly from adjacent neighborhoods. It's a pretty solid middle class neighborhood that is well-suited for outdoors types and for people who like to stroll through the neighborhood on good weather days. Restaurants are great, with awesome organic and vegan options. People are very friendly, if not overly politically correct. The neighborhood association is very active and protective - there's currently a campaign to stop a Whole Foods from moving in. There have been some isolated violent crimes recently - but it's hard to escape that in any metro area.
With all that being said, the commute to Harvard's main campus in Cambridge is not ideal. Though, I'm not sure how different it would be from lower Allston. If you'll be based at HSPH in the Longwood area then JP is perfect...
I'm going to miss it (