Many of the answers you seek can be found on The Villages website:
The Villages(R) Florida: Active Retirement Living at Its Best
You should check there first.
Also - palms aren't technically trees, they're just plants. So yes you can have them removed without prior approval. Other trees depend on the reason for taking them down, and what kind of tree they are. Oak trees, bald cypress, and Florida pines (among others) are considered protected trees in the state of Florida, so unless you have a dead or diseased one in your yard, you can't remove it. You /can/ prune protected trees if it's light pruning, but any severe or "stripping" pruning of protected trees has to get a sign-off from the county.
If the tree isn't a protected species I believe you just need to make sure an arborist says that removing it is an acceptable course of action. You'd need to double-check the law on that, there was some change regarding non-protected trees and tree removal in the law in the last few years.