I have lived here for close to 5 years. A basic apartment fairly close to the St George campus (as in 15-20 minute subway ride) will be approx 1000 plus utilities. Specific neighbourhoods to look for places would be Riverdale, Cabbagetown, and the Annex. Perhaps you'd like to try looking on Craigslist to get a sense of what is available. I'd also consider living a 30-45 minute TTC ride away (Yonge and Eglinton area or King West).
I certainly haven't had any run-ins with bedbugs, rodents etc. Toronto is a gorgeous (and comparatively, extremely clean) city.