I would check with ARC before doing anything. I vaguely remember something about Live Oaks having some special status. Not sure if it's a Villages thing or State. Keep in mind the appearance of the tree after significant trimming may be much worse than leaving it. Personally, I wouldn't touch it unless it was a danger or problem.

I would love to have an oak tree providing shade for our home.

We had an issue with the tree behind us from the neighbors oak tree. We cut some of the huge branches that were almost to our lani screens. You can do that without getting in trouble. We already looked into this… before too long we’ll be cutting the other huge limb….

General Rule
 

I can't speak to Florida law. However, this statement is in line with what I was taught in law school. In addition to being allowed to cut the branches and roots back to a vertical plane on the property line, the neighbor is not liable if the trimming results in the death of the tree.

The general rule is that the owner of a tree or a branch which falls on the neighbor's property is not liable for the damage, absent negligence. Examples of negligence which would result in liability for a fallen tree or branch include the failure to trim branches; the improper trimming of branches; and, the failure to remove a dead or dying tree. Although rather far-fetched, the owner would also be liable for planting a tree that, although otherwise appearing to be healthy, was unsuited to the soil or climate conditions and failed to establish a root system that would allow it to withstand the prevailing winds.

Perhaps, the last example of potential liability is not that far-fetched. Our son lives in a town on the seashore south of Boston which is subject to high winds. Years ago, a particular type of tree was extensively planted on the hillsides for erosion control as these trees developed an extensive root system. However, as the trees grew to their full height, it became apparent that the trees would suddenly break-off at mid-height in a windstorm and not bend. Property owners have been warned to "top" these trees or face liability if a neighbor's house is damaged by a broken tree.

True about the center of the state, but NASA stated that this summer is the warmest on record. Therefore. hurricanes could become stronger each coming year and might reach into the center of the state.

They have benefits but also come with problems like dropping leaves into gutters all year and encouraging mold.

Trees
 

Florida law says your neighbor is not responsible for those over hanging limbs and also states you can have them pruned at your expense to the property line without his consent. It also says anyone he hires to prune needs your permission to come onto your property to prune, and don't be surprised if he asks you to pay for pruning what's over your property.

Great shade, but thousands(if not millions) of acorns.

I have The Villages trees in my backyard yard and branches were over my roof, it was my responsibility and cost to have them cut down.