What else can we do to stop this behavior? 
Rubbing his chest or belly to calm him down isn't going to help. It's attention, and any attention is good attention. When he gets mouthy, your best option is to immediately end play. Close a door, step over a baby gate, whatever, just make sure fun ends when teeth meet flesh. You MUST physically remove yourself. It is not necessary to grab his collar or usher him outside. You should also encourage him to bite his toys and not you.
The growling and nipping when he steals food items is likely resource guarding. A good resource for that is Mine! by Jean Donaldson. You should look up the "trade game", as well. Management is another important factor. I know it's difficult with children, but you should try your hardest to prevent the dog's access to food he isn't supposed to have. You can do that by using baby gates (to keep out of the kitchen) or teaching the dog a "place" command. For example, when children have food in their hands as they often do, the dog should go to his bed and stay there until released.
The dog should eat his meals or have high value chews in a private place like a crate or separate room, as well, so he never feels threatened.
Although you have a good start with learning the common commands, you do need to proof them, preferably in situations that you have set up so you can control them. You might try looking up Kikopup on YouTube. She is a really great positive dog trainer and seems to have a video for just about anything you can think of.
The growling and nipping when he steals food items is likely resource guarding. A good resource for that is Mine! by Jean Donaldson. You should look up the "trade game", as well. Management is another important factor. I know it's difficult with children, but you should try your hardest to prevent the dog's access to food he isn't supposed to have. You can do that by using baby gates (to keep out of the kitchen) or teaching the dog a "place" command. For example, when children have food in their hands as they often do, the dog should go to his bed and stay there until released.
The dog should eat his meals or have high value chews in a private place like a crate or separate room, as well, so he never feels threatened.
Although you have a good start with learning the common commands, you do need to proof them, preferably in situations that you have set up so you can control them. You might try looking up Kikopup on YouTube. She is a really great positive dog trainer and seems to have a video for just about anything you can think of.
