OK, so that behavior is pretty common in young lab puppies. Not desirable, obviously, but they typically outgrow it with a LOT of reinforcement.Those nice calm labs you see in movies or on TV are not puppies, they've outgrown the biting, nipping, jumping baby puppy phase. There are some behaviors you can teach your children that might help in the meantime. Your older kids for sure should be able to do this, the younger ones will learn but any interaction with the puppy and your littles will need to be closely supervised until puppy gets older and more under control.
Use the SAFE Program to learn about dog bite prevention
I wouldn't feel too bad about putting the puppy in her gated area for rests frequently through out the day. My puppies have always been on a pretty strict wake/rest/feeding schedule, just like I would a baby/toddler human. I don't know what I think about having the puppy on the tie out with the kids running around, I'd probably keep the puppy inside so she doesn't get so worked up watching the kids and take her out for play time separately or with your older kids who will learn not to reinforce her jumping, nipping behavior. Our older boy was bad about running at us and jumping, mouth open. I learned to spin away from him when he did this but encourage him to keep running to help get the excessive energy out.
Shelley posed while I was writing this. I agree with everything she said.