Moving to Montezuma or Cabuya

Hello, and welcome!
We didn't deal with bringing a pet, but plenty of others have - flight and immigration/custom are no real problem as long as you have all the proper shots and documentation (w/ proper timing), and bring the dog on the same flight.  Many smaller hotels (especially B&Bs) are pet-friendly. 
Dogs are not allowed on buses, so you will need to consider how you will travel around the country, and whether you will bring your dog with you, or get a pet-sitter or use a kennel (once you are settled).
re banks:  Citibank (and other "name brand" banks) here are not branches, but rather franchises.  You do not have access to your California account directly.  So there is no real advantage to sticking with the same bank name.  In fact, there is a slight disadvantage of the private banks compared to the public banks, such as Banco Nacional - the public banks are insured by the CR government, while the private banks may or may not be covered for collapse by their parent bank.
If you are looking into immigrating, as opposed to snowbirding or perpetual tourism, check out the requirements here: www.arcr.net.
good luck!
Julie