Woohooooo, that's my city! Actually, I live in Mt. Pleasant by the Isle of Palms connector.
James Island County Park has a dog park, off-leash area...I didn't know there was one on IOP but I'll definitely check it out.
Hmm as for pet friendly places, there are many! Check out our local magazine: "Lowcountry Dog"
http://www.lowcountrydog.com/page/page/1652269.htm
and it has a list of dog-friendly places/stores/etc.:
http://www.lowcountrydog.com/page/page/1652269.htm