Is he neutered? I forgot to mention that in my other answer. Neutering along with taking over as top dog should settle him down. The only answer to getting loose is to be very careful about keeping the gate closed. With 2 children, that can be difficult, but remember, I said not to leave the dog and children alone. You can likely give him an outlet for his energy and eliminate the barking by playing fetch. The discipline of giving you the ball or stick will help with the top dog thing. Having a dog is a time consuming business. You can't just put some of them outside and let them take care of themselves. Think of him as a third child. When you have had all of fetch you can, if he still wants to run and bark, take him inside. If you have to, crate him. Other dogs may not be as bad as the young Labs I am plagued with. Still your house and dog
Will be much safer with the dog in a crate when you are away. The dog may be
Happier in its den than loose in the house. It relaxes, it feels safe in its
Den. It rests, the body slows down reducing the need for water and relieving
Its self. Dogs that have been crated all along do very well. Many of them
Will rest in their crates even when the door is open. I think the plastic
Ones give the dog more of a safe, enclosed den feeling. They are harder for
Dogs to open too. Metal ones can be put in a corner or covered with
Something the dog can't pull in and chew. Select a crate just big enough for the full grown dog to stretch out in.

Leave it some toys. Perhaps a Kong filled with peanut butter. Don't leave
Anything in the crate the dog might chew up. It will do fine without even any
Bedding. You will come home to a safe dog and a house you can enjoy.

A dog that has not been crated since it was little, make take some work.
Start out just putting its toys and treats in the crate. Praise it for going
In. If you have been able to trust it with any bedding, put that in the crate.
Feed it in the crate. This is also an easy way to maintain order at feeding
Time for more than one dog. It will also protect your children if he wants to protect his food dish from them. There are ways of dealing with that too, but it can go until you solve some other problems.

Use the crate only when you must I have a short length of chain fastened to
My computer desk. With more mats around the chair mat, this gives a place
Where the puppy is with me, but about as controlled as in the crate, while
Still allowing me to get some work done, or at least answer AskMe questions.

Note, I decided to combine the 2 questions on the same dog. It will make all the information easier to read.