The best deals on rentals will probably be near any of the colleges. My two-bedroom, 1 bathroom apartment that was 700-something sq. feet, was only $575/month - and that included utilities, cable tv, and internet/wifi; I lived just a block up from the northern edge of the main bulk of MSU.
So, yeah, I would definitely look into houses that are near one of the colleges, in terms of pricing. As for dogs... not sure; I never had a pet in Springfield, and I was in an apt, not a house. Thankfully, the limit for number of dogs owned in Springfield is four, so you're good there.