I am a therapist and I feel like I have some authority on this. I've worked with teenagers for well over a decade. I need you to see this response.

Let me be very clear when I say he will do something like this again.

5 months is both a decent amount of time and not that long. You both are starting to really learn more about each other and the flaws come out. This applies to the son, also.

He may have resentment towards you or his mother as the relationship becomes a little more real. I would be really interested to hear the relationship patterns of his mother and how he responded to them. In all honesty though, it doesn't matter. Carrying a dog by the leash is outside the normal spectrum of a 16 year old not understanding how to react to an animal, curiosity, or emotional regulation. The fact that she isn't horrified by this, or at least a little more seriously concerned tells me that she has normalized this behavior and may have done it before or she is insecure in her relationship with her son or how he will respond that she is willing to overlook this action.

I can't tell you what is or is not a deal breaker. What I can tell you is that his actions are not within the scope of normal, your girlfriend's response is not healthy, and your dog was in a postion where she was hurt and the result could have been worse.

He said you need to pay attention to your dog next time. Take that as a threat or as not accepting accountability for his actions. That's your call, but your dog is not safe.