Battersea is great - full of young, single professionals, great bars and restaurants, and Battersea Park will be great for the dogs.
Otherwise, there is an area that estate agents call 'Between the Commons', which is similar, but a little more green and further fromt he river.
Although Wandsworth isn't on the tube, it is great for overground train services and buses, and you can be in central london within about 10 minutes from Battersea.
Vicky