Are the hiking/backpacking trails open on Saturdays and Sundays?

Yes they are

The palmetto trail stretches across a good portion of SC, with many dock/bridge waterway crossings, welcome to fishing aswell. It's welcome to backpack hiking, bicycling, an horseback. No motorized vehicles permitted on hiking trails although wadmalaw Trails have accessibility to dirt biking an 4 wheelers, you can Google many attraction areas for many areas..