Oh yes, brother dogs. They are so much of an adorable and frustrating handful!!!
Our first two dogs, Arby and Rax, were brothers. They were the best of friends and the worst of enemies, playing one moment and fighting the next. They also had this terrible habit of escaping. We built taller fences, set logs up against the bottom, put them on a lead while outside, and used shocker collars. They didn't stop jumping that fence until they were so old their hips didn't let them.
Our second pair of dogs were also brothers, and they are far more prone to running away when together then if just one is outside. However, they did better then the first two because they did not just have themselves but two other dogs to follow about and bond too.
Anyways, sibling dogs are so wonderful in their bond but far more difficult to train because their first love is for each other and they can become more disobedient because of it.
You mentioned that Levi was well behaved and Hunter was the hyper one. What breed are they? They sound large, and as they start reaching those teen years trouble starts brewing. They become easily bored, want to search out girls and new territory (even if fixed), and test their boundaries on what they can get away with. This is when they need the most exercise, training, understanding and discipline.
The first thing you want to avoid is getting scared, angry, or tired of the dogs. Dogs can sense this and use it to their advantage in any way they can. They don't bond well to people who distrust them, they need someone to be willing to make the commitment to get to know them.
Also, teen dogs thrive on a schedule. When they eat, when they are locked up for the night, and when they are walked. They have fast minds and energy-filled bodies that need both mental and physical stimulation. They need a lot of time given to them and actually, a child above the age of 12 may be able to help with many chores. However, younger children often cannot handle strong, easily excited dogs and should always be supervise and helped to understand their dog and the dog guided in understanding how to respect and be gentle with the child.
Steps towards helping a teen dog pair: Walks are number one. Walks, walks and more walks. No matter what time of day, a walk, jog, hike, anything that makes a dog go at a good pace and keep it up will help in SOOO many ways. After all, dogs ancestors, the wolf, walk or run for miles and miles as a pack looking for prey.
Two. When training or walking a dog, sometimes it helps to do it alone. Don't feel bad if one dogs is left behind as long as it gets its turn. A lone dog focuses on its owners energy and commands above the desire of its brother and will better behave and likely enjoy the walk better. Also, you can put all your own focus and energy into one dog, instead of trying to control two, which will only get your frustrated.
Three. Mental simulation is also very important. Tracking scents, meeting other dogs, being taught agility or tricks, being given puzzle treats (kong, ect), playing with toys, it all helps them use their mind like a wolf would have to while hunting or like a dog does when doing a task like herding or retrieving. Whatever your dogs breed(s) are will help you identify what they will most want to do: track, hunt, herd, guard, retrieve, point, flush, draft, ect.
Four. Daily spend time with each dog, petting them, praising them, and teaching them some very important manners also. Do not allow by any means biting or mouthing, because of children in the house this is dangerous and must be avoided. Jumping is also a no. However, yelling at a teen dogs tends not to get great results. Instead, be calm, firm, and simple. Tell them no and either take their collar and have them look at you, or tap them on the neck to get their attention (you aren't hitting them, just getting them to switch their focus to you). Tell them no and stop whatever you are doing until they except it. Keep this up and be consistent, even if it is hard.
Teen dogs love to spend time with their family. Being out-back may be a good answer if they are too hyper or destructive, but steps towards getting them indoors is a wonderful idea because the more they are a part of your pack (family) the more they will learn how to live with and work with you family and your rules and boundaries.
Again, before introducing them to inside walk them both. Then take one, when they are calm, and let them inside when either no one else is there or else only one or two calm individuals who understand not to get the dog excited.
Let the pup run around and sniff, perhaps even jump on the furniture for a bit as it gets its bearings. Then call it to you and offer a treat, perhaps by letting them sit. After this you can take a seat on the floor or couch and remaining calm begin telling them what the rules are. Perhaps you don't want them in a certain room, on the table, in the trash or on furniture. Make it clear what it is you want and understand that it will take some time for them to grasp it. Covered trash cans, gates or closed doors, and making sure the children of the house understand the rules and know how to enforce them with no danger to themselves are all ways to make the training go about more easily.
Once each dog is good on its own you can introduce them with each other in the house. Again, after a long walk and when they are calm and the house-hold is calm will be best. To avoid temptation, make sure there is no food, chewables or breakables in easy access while they are learning. Also, make sure the dogs fully respect the children of the house as their leaders and that they do not growl at, bowl over, nip, or do any other such behaviors towards them. The kids must be confident in their role as the dogs leader and not run away or hide when they see the dogs, which will just lead to the dogs being confused and making the frightening behaviors worse.
The issue with jumping the fence is a very hard one. I do not know what it looks like, which makes this more difficult, but there are some things you can try to do about it. Building up a taller fence of making a cover are some options that may or may not be expensive, depending on if you can find some way to build up the fence taller for a low amount of money. Dogs that are bored, anxious, or have a build up of energy they can't get out (especially bad in large working dogs and teen dogs) are prone to destructive behavior and escaping. To help them feel more comfortable in their pen make sure they have a warm, dry place to stay, clean water, cool shade, dirt to dig in, and toys to chew on. Putting up fencing or boards around the tree may help him stop flinging himself too.
Levi sounds like a dog who would more easily integrate inside. However, take care that you don't favor him or Hunter may begin to act worse then before. Hunter is the one who seems to be highly intelligent but also obsessive. Dogs who obsess over something (escaping, digging, ect) are very stubborn and intelligent and very good at focusing on a task. Getting him some sort of job will help him a lot and is a great time for human and owner to bond. Agility is a wonderful way to get a hyper dog to exercise mind and body. Start simple and build up. You can also go for hikes in the woods, teach fun tricks like spin, jump, sit and down (some hyper dogs can't sit still that long) or so many other fun ideas like frisbee, swimming, ect. It
In the end, if you cannot handle the dogs do to their energy, aggression, escaping habits, or the high amount of money it may take to build them their run, do not feel guilty for finding them a new home. Instead, realize that you are doing yourself and the dogs a favor by relieving the stress on both of you. You can adopt them out together, as they are bonded and it is stressful to be placed apart. Find them the best home you can, one with space and understanding owners who know the work that must go into them.
Many people want to say only a bad owner has to find a new home for their dogs, but we have rehomed some of our pets for various reasons. They didn't get along with our previous animals, they had health issues we couldn't afford to care for, we couldn't afford to feed them at the time, they were not getting proper exercise, they caused allergies, or they were causing too much stress on our family. Again, finding them the best homes and knowing that you gave them that is a wonderful thing to do and not something to be ashamed of.
I hope this helps you and your friends and family with your two dogs. Wishing you the best of luck!