You probably want to peruse the
Raleigh/Durham/Chapel Hill forum for areas near Raleigh. There are TONS of threads, but it would be helpful to provide a little more about what you're looking for--urban, suburban or rural, how many BR, where you'd be commuting to, etc?
There isn't any city to "avoid at all costs", though all cities have better areas and areas you would want to steer away from (like anywhere else). It really depends on what you're looking for and what you want to avoid.
If you have a detached house, there aren't really any rules about dogs, though there might be rules about not letting them run unleashed without a fence. That kind of thing would probably be more of a HOA thing than a "city code".
Read the many questions people have already asked, and elaborate more on specifics. Your question "where should I live?" is still pretty general.
Have you visited the area? That's a first step, of course--once you've seen the area and can narrow down your search, is when this board becomes most helpful.