1) There are many different valid ways of exercising. Walking a dog is one of them, and if you had ever owned a dog you would know that they don’t need to do their business every 10 minutes.
2) A properly trained dog will not make a mess in your house. If you’re worried about shedding, get a short-haired dog and keep it properly groomed.
3) Feminism is not the same as egalitarianism. Feminist theory does not, as is popularly and mistakenly believed, advocate that women are the SAME as men. I believe in Jewish values of gender roles, but that does not mean that I cannot also be a feminist.