Work? - I don't think so. You need a Permit to work in Belize and this is very hard to come by... Ask any of the many ex-pats who have been living there for over a year and are still waiting.
Regarding a house for long term rental - best to check your options when you get there. I am sure you can find, but the prices range depending on location, among other things.
Good luck!
Cheers,
Diane