I'd choose I'll leave instead of I'll go out. And behave instead of be good. If you don't behave, I'll leave. I think we need the context to give a better feedback. I hope I helped you.

Going out, generally means leaving the house. So no, it wouldn't sound right if you just mean going from one room to another. I can't think of a natural way of announcing that you are about to move from one room in the house to another - it seems an odd thing to do. If you have had enough of a conversation with a person and you want to move away from them - whether to another room or just another part of the same room, you might say something like "I have had enough of this", and walk away.

I think it's better to use " l'll leave the room "

I think going out usually means leaving the house. "I'm just going to the kitchen/bathroom", and yes, for the child example: If you don't behave! / If you're not good, I'll leave!!"