Dogs are another story. There are some breeds that are restricted because of insurance. You may need to negotiate with a landlord for the pet allowance and perhaps pay an additional month premium or a security deposit.
No sense beating about the bush. You are unlikely to find what you are looking for in Boston, Cambridge or Somerville for what you are prepared to spend. My best suggestion would be to look at a more outlying area, and if your campus is accessible to public transportation, to use one of the MBTA commuter trains to access school. It will mean budgeting time to accomplish your commute, but with some planning it should be possible. I would suggest either the Lowell or Haverhill lines which commute to North Station, or the Worcester Line which goes to South Station (which in turn connects to the Red Line). Here's a link to the map and schedules. http://www.mbta.com/schedules_and_maps/rail/
I feel fairly certain about the information I'm giving you, as a young friend of ours, just starting off, has been looking for a rental for $800/month. He has been finding shared apartments (with room mates) and the occasional basement studio -- not altogether a satisfying result so far.
I wish you the best in your search. Try some of the areas 25-45 minutes from Boston by rail, and I would think you'd do better. If the school you will be going to is not accessible to public transportation, it may have a shuttle service to the nearest subway or rail station, as many campuses do. If all else fails, you may have no choice but to endure a commute of a half hour or more.