I don't know what the time limit is on being inside Canada.
For the time you are in Canada, you can freely travel amongst all provinces and territories, yes.
While you're permitted here, I'm pretty sure you're free to become a regular tenant. Like Cookie78 said, though, areas with very low vacancy rates (and there are lots of those) will make it harder to find a six-month term. You might look for housesits or sublets. But with dogs, housesits, sublets, and regular rentals are all harder to come by.
Some landlords are totally cool re: ER, sabbaticals, students, etc. Some only want people who go to work 40 hours per week. Again, the more vacancies in a town, the higher your chances; the fewer vacancies...your chances plummet, like with the many homeless professionals.
What about an RV? A home for you and your dogs, can drive to the available spots or boondock...