He shouldn't have brought a puppy home without talking to you first, but it is done and to make him give him up now would be like him asking you to give up your cat. Insist he takes care of the pooper scooping and bathing him. When he is old enough, have him fixed, it will settle him down and keep him from being as agressive. You can start training him now. Cut up a piece of bologna or hotdog or use liver treats. Make sure there isn't any distractions so do it during the time your children are sleeping. Hold a treat in one hand, swish it across his nose so he knows to watch the hand and say "Watch me" as you bring the treat up to your nose. Work on this as much as possible. Give him the treat only when he is sitting and looking you in the eye. Once he has mastered this use the Watch me command whenever he is being distructful. Remove the objects he is chewing on and give him one of his own toys or chew bone. Make sure he has plenty of chew toys that squeak because that is his reward for chewing on them and not the kids toys. Make sure when you take one of the kids toys away you say "NO" then give him his. They learn to only chew on the ones with their smell that way.
Potty training seems to be going good for you if there is only one accident a day. Watch to see if it is the same time every day and make sure you take him out at that time. Another thing you could do is hang a bell by the door low enough for him to hit it with his paw. Each time you open the door to let him out, ring the bell. You might show him that he can ring the bell by taking his paw and hitting it saying "get it" then try telling him "get it" and see if he will try. He will soon associate the bell with going out to go outside and will ring the bell when he needs to go out. This works great with my dog since when she would scratch on the door, if we were in any other part of the house we wouldn't hear her. The bell I use is a cowbell that lets out a loud ring so I can hear it no matter where I am in the house. Only problem is she has learned if she wants my attention she will ring the bell to get me to stop what I am doing and let her out even if she doesn't really want out. When she starts doing this I just put the bell up for a bit.
Once you have the watch me command down, you can work on the "come" command which is one of the most important commands since it keeps them from running out of the yard or going places they shouldn't. I take a treat and like the watch me command I let her know I have it then bring it from the side to my chest saying come. When she comes I give her the treat. You can also do this with the lay command. Make him lay while you are saying "Lay" or "down" and that might mean that you have to pull his front feet out to show him you expect him to lay. When he is laying give him the treat. When he has this down, use the lay command or down command whenever he starts to jump on someone. One more thing you need to do is make sure that he knows your children are alpha or boss. Have him lay and let the kids one at a time pet him, scratch his belly and mess with his feet. Make sure he is submissive when you do this. Always feed your children before you feed him if they are awake. Another thing I did when my dog was a puppy was use a rolled up newspaper (just one section, not the whole paper) to hit on my hand for a loud clap if she did something she shouldn't, like get up on the table. I called it puppy paddling (sometimes I would swat her on the hiney with it light enough not to hurt but enough to have the paper make a noise). Wasn't long if she was caught chewing on something she shouldn't I would say "need a puppy paddling?" and she would stop. She is now going on 10 years old and there isn't a dog around that is as fun and amusing as she is. She has different tricks, the favorite is when I hold my fingers like a gun at her and say "stick em up" and she will stand on her hind legs with front paws straight up in the air. I say "bang" and she lays down and rolls onto her back playing dead. She is a cockerspaniel and is so smart. When we first got her everytime we would say "is someone here?" she would let out one bark and go to the door. My husband thought this was really funny so he would do it over and over. She finally learned that she could go to the steps in the livingroom and look in the mirror on the china hutch and see the glass door to see if someone was at the door. Then when my husband would be watching tv she would bark once and head down to the door. He would get up and say "oh someone must be here" and go look to see no one was there. He sit back down and she would do it again. This went on for a while and I sit back laughing because I knew she had turned his game around on him. They really do have a personality like a child.
Dogs are wonderful pets and yes they are high maintainence but they are well worth it if they are trained and behave in a proper manner. It might take work when they are small, but once they get it they are such a pleasure to have around.