Installing a pool will probably be ok but it depends on the HOA for the neighborhood. It's not hard to get the HOA restrictions for any given neighborhood.
Property taxes would be about $1608 on a $400k owner occupied house in Mt Pleasant. You can find a property tax calculator for Charleston County here:
Auditor's Office
Flood insurance is not required for properties that aren't in a flood zone. A good bit of Charleston National is actually not in a flood zone though the others you mentioned are mostly in a flood zone at this time. FEMA is currently redrawing the flood zone maps so that might be changed within a year or two. There is another thread on that topic here on the forum. The price for flood insurance can vary greatly and needs to be looked at on a property by property basis.
As for removing a tree, it depends on the tree. A small tree; no problem. A Pine will probably be easy to get permission for. A large Live Oak; Probably not unless it's diseased or posing a danger of some kind.
Hurricanes are a fact of life here though in the 33 or so years I've been here, the only one that has had any serious impact was Hugo in '89. Of course some properties are more vulnerable than others but I wouldn't say Mt Pleasant in general is any worse than any other close-to-the-coast area. Within subdivisions there might be individual streets that were impacted by flooding, or even individual houses, depending on water run-off. Whole neighborhoods at a time weren't flooded out.
For the kids sports... As Qtopplings said, there seems to be every sport represented around here and especially in Mt Pleasant. They would use school and public parks.
It's good that you have visited and know the area a bit. Come again with an eye towards living here if you can. :-)