Dexter Going to Heaven

So this is a sad post. We got a dog in January at the pound, a Jack Russell. When we brought him home he was very mellow and then he starting snotting everywhere, he had a cold. He would also make wheezing noises, so I took him to the Vet, he had kennel cough. We cured that by shoving meds down him for two weeks. So he has been a relatively good dog, besides not being potty trained and he wasn't neutered when we got him. If you know anything about male dogs, if they are not neutered by one to one and a half everything they do is now pretty embedded into them, its very hard to break their habits. He did pretty good. He has this thing if the door opens he will just take off running and we have to chase him down and find him. He has run away a few times, its not fun. A few weeks ago one of Tabatha's friends came to play. They went outside and Dexter decided he wanted to jump on the girl and bit her coat, maybe even her. I don't know what it is with him and the kids outside together but its just not a good thing, he is too hyper.
The other day he got out and we couldn't find him for a while, finally the secretary in our ward found him and brought him home. Later that night I got a call, from someone in our ward, that Dexter had bitten his daughter. He didn't break skin or anything, but he wanted to inform us. Today I went to go get the mail, I take Dexter with me and run the whole way, to get him some exercise. On the way back we walked by a lady with a Shitzu, Dexter went up and just bit the dog and wouldn't let go. I had to drop the mail and pry his jaws off. Thank goodness he didn't do any harm, the other dog didn't even yelp, but still SCARY! I finally can not take the guilt and liability of this any longer. The previous owners of this dog and caused him pain where he feels what he is doing in necessary. I a one to think I can always fix things, but I cannot fix this dog, I have tried. I reported him to Davis County and someone is coming to get him tomorrow. I really don't know if they will put him down or not, but he has to go. I don't think he would hurt my children, but who is to tell, I cant take that risk after what I have seen over the past week. I feel really sad but I know its the right thing to do.