Freedom Park

No

So the sidewalks around the Carter Center and the Carter Center itself is mostly a seperate entity from Freedom Park and the Stone Mountain PATH. The Carter Center exists withing the east and westbound lanes of Freedom Parkway, whereas Freedom Park occupies mostly just the excess right of way surrounding Freedom Pkwy. There is street parking for Freedom Park all around the O4W/Atkins/Inman/MLK neighborhoods, as well as one small parking lot at the Dog Park. You can access the Stone Mountain Trail PATH via dozens of access points between Stone Mountain and Centennial Olympic Park. My best suggestion is to find some street parking, or take transit, and spend several hours exploring the area, it's easier to find out by exploring it than it is to read about it, because there's a lot going on in this area. Best of luck.

They do have them and I have seen them on nites and early mornings. I'm not sure what the schedule is tho.

Yup

No fees it's free lake for fishing...

Yes it does. A regular and a splash pad also

I've seen plenty of people laying around with blankets, playing frisbee, and having what looks like picnic items (baskets, containers, etc.) There is enough green space to not be near the congested roads (I recommend).