My immediate thought here is Adams Morgan. It's certainly walkable (two grocery stores in the neighborhood, lots of restaurants, bars and shops, etc.) and has Walter Pierce Park (which includes a dog park), Kalorama Park, and is close to Meridian Hill Park. It's also tree-lined and leafy on most of the residential streets. Plus, the 42/43 bus line runs right along Columbia Road (which cuts through Adams Morgan) and goes directly to Farragut Square — probably about 20 minutes or so of commute time.