What if someone went inside our gate (without permission or unknowingly), and got bitten by our dogs, who's responsible? Sometimes our neighbors' kids play on the street in front of our house, and sometimes their ball gets over the bakod. They would courageously (or stupidly) get the ball themselves instead of asking us to get it for them. We're worried that someday one of them would get bitten. What are the possible actions for both sides?

The tresspasser whether minor or of age is liable for the injury(dog bite)sufferred. This is a case of damnum absque injuria (There is damage caused but no legal injury enforceable in court)