Re dogs: Goose Green is fully fenced - plenty of dogs.
While Peckham Rye isn't fully fenced, it is very large - so if your dogs like to run, but will come back when called they can get a lot of runaround there (again plenty of dogs), and as a Melbournite - you can barrack for the SE London Giants aussie rules team there too. As an aussie now in ED a few years, love the area.