Can only speak for ORP.
No restriction on the number of domestic pets. Fences are permited but must meet Architectural Review Board (ARB) guidelines (would be part of the landscape design of your home).
As far as the ARB goes, our house plans will be reviewed in the next month so I can tell you more after that.
If you buy a resale you will avoid the ARB step and probably save a bunch of money.
If you decide to build, it would help a great deal to work with a designer who is familiar with the ARB design standards, that will make that step much faster.
Here is a partial list of the active groups at ORP-
Ocean Ridge Plantation Real Estate | Ocean Ridge Plantation Homes for Sale | Ocean Ridge Plantation Builders | Sunset Beach North Carolina
Take copious notes on each of your tours (to keep the communities straight) and have a great time in Brunswick Co.