I would like to offer more services but for the moment I can only add House Sitting to the list. Would I be able to bring my dog with me to the houses I stay in with the client's permission? What would be the requirements for this?
The answer to that question is going to vary from one client to another.
Before you ask the client, first you need to ask yourself, how would your dog do interacting with a client's dog (would your dog be generous sharing your company, does your dog seem like it would like the company of this other dog, do you think the other dog would enjoy the company of yours, and would your dog be a good house guest). If the answers are yes, yes, yes, and yes then ask the client about this prior to the meet & greet. If the client agrees, you'll need to bring your dog to the meet & greet and introduce the dogs. *If you haven't done a lot of this, read up for tips on best ways to introduce dogs prior to meeting (outside/neutral turf, paying close attention to their body dynamics, and start with a short parallel walk together while you and the client chat. Before you get there you may even have a 1-on-1 talk with your dog about being on best behavior, meeting new friends.)
At the meet&greet, you and the client can agree on if you think it's best or not.