What is the best neighborhood for an urban family?

My husband and I are in our 40's and relocating to Minneapolis from New York City with our 5 year old son. We would like to find a neighborhood where we can walk to restaurants and stores and that has good access to parks and lakes, plenty of kids, and some diversity. We prefer things a little gritty rather than fancy, but want to be in a safe fun area. Neighborhoods we are considering are East Harriet, Kingfield, Lyndale, and (in St. Paul) Macalaster Groveland or Summit Hill. Some of these are zoned to Lyndale Elementary, which has low test scores, but that is not necessarily a deal breaker for us if it is otherwise a good school. Help! Where would you recommend we look?