Belize is like almost anywhere else in respect to money -- you can live very cheaply (well under US$1000 a month for two for a pleasant Belizean-style life ... or not (US$5000 to $7500 a month and up for a U.S.-style way of living in a high-cost area, with A/C, car, imported food and booze, lots of dining out).
It really depends on where you live -- Belize City, San Pedro and PG being the most expensive areas, anywhere rural and Corozal Town being the least expensive. It also depends on your lifestyle.
But for various reasons -- the small size and inefficiency of the local market, relatively high labor costs, high cost of imported good, among other things -- Belize is not by any means the cheapest place to live in the region.
Having dogs shouldn't pose much of a problem for rentals, unless you are looking at a condo or smal apartment in San Pedro.