This program does not entitle you to be employed in Belize. Both are costly and approval is uncertain. Check with a local attorney for details. General Work Permits must be applied for by your prospective employer. That employer will have to prove that they have been unable to find a current local resident to fill the position before a permit will be issued.
Timing varies with workload of the department.
You generally get straight, helpful answers from the people doing the work. No quarantine, no visa. And yes, we have good vets here. American Airlines has particularly good and caring pet-travel programs. These fellows don't charge much for their services, and in the end will save you enormous amounts of time, energy and money.