Hi Carollee. That's a lot of questions for one post, you can probably find the answers to many of these individually and then ask more specific questions later. Many of the answers are going to be subjective, based on your likes and needs. Here's my attempt at answering a few of them.
Can I bring my car into Belize? Yes, and you'll pay an import duty on it. But you can also buy one here. Many towns are walkable and you don't need a car. I've heard that San Pedro and on many of the islands, golf carts are the main mode of transportation.
Do I have to keep my US passport valid after I move? Of course. And you'll need to renew your visa periodically. I'm an "extended stay tourist" so I renew monthly, I think the folks on QRP (retirement) can do 3 months at a time, but I'm not positive.
Can I bring pets into Belize? Yes. Look up "Belize BAHA." You'll need to have your pets looked at by an appropriate vet and fill out the paperwork. If they're small you can bring them in the cabin. If they're not, you'll probably have to wait because I'm not sure any of the airlines are allowing that at the moment and the land borders are closed. There are also individuals who take care of planning this for you as a consulting service.
Where can I rent a small house near the beach? In practically any beach town.
Where is the best small town to live in Belize? Depends on what you like. Sounds like you need to visit a few places and see what they are like. Have you been here? Maybe plan on doing a longer stay and going to a few places you are considering and staying a few weeks at each. Kind of a fact-finding mission.
Can I keep receiving SS from the US while living expat? I have no idea.
Is there internet service? Cell phone service? Yes and Yes.
Where is the expat community?
There are a lot of expat communities in various places in Belize. One thing you'll hear a lot about is not expecting to come here and live a North American lifestyle though. It really sounds like you need to plan a visit to be honest. Most come here for a more simple life. Trying to bring the life that you are living at home to Belize is bound to be more difficult. You will need to learn to be patient (even if you already think you are) and expect that things aren't always going to go the way you want them to go.