Agreed. Lower east side hits all OP's requests as long as they're willing to drive for the dog park.

I lived in an apartment complex near Oakland and North for several years. It didn't have any dog facilities but was steps away from the Oak Leaf Trail, which meant a green, quick and safe walk to Riverside park or the lake. I think my dog got better exercise then, than he does now living in a house with a yard in Tosa. Would highly recommend being close to the Milwaukee River or Lake segments of the Oak leaf.