Pisgah National Forest is only 15 minutes away and has many camping spots.

Yes, but they must be on a leash. Source: Park website: Tab across the top marked "Explore the Forest" and the pull down menu " about the Forest" scroll down to the bottom of the page to the area marked "important forest rules".

Don’t know for sure but the visitor center is closer I think

Sure, but my recommendation is to get there and go to the ranger station for advice and maps / info to where best to picnic for what you would like to see.

The covered bridge is pedestrian only but the vehicle road is quite close.

There were families there with strollers when I have visited. There is gravel so as long as your stroller can handle that you should be fine. There are steps so you would have to leave the stroller at the top if you chose to go to the lower viewing area. There are plenty of places to view the all the falls, the river, the covered bridge and nature without using the steps. Enjoy!

I would say triple falls are optimum for kids to play around and do a little bit of hike as well. Pools are very expansive their with flat rocks to have little bit of picnic as well.

DuPont State Forest and Pisgah National Forest are less than an hour from Asheville. Both have waterfalls.

The water is very shallow at Hooker Falls, so you shouldn't have a problem unless you get close to the center of the falls where it is a little deeper. A life vest couldn't hurt, but probably not necessary. You can climb up the rocks and get behind the waterfall, but that could be dangerous. While we were there, there were 2 rangers watching and never told anyone to stay off the face of the Falls. There are signs to stay off of the top of the Falls.

It depends on if they are great swimmers , When I was there we where just playing under the water fall , but there was a bunch of kids swimming , it's not that deep at all , be safe have fun !!!