D..
Your heart's in the right place, but here's my advice to you.
You moved in 6 months ago. You aren't this man's wife. But you are taking on a really big responsibility. Perhaps too much of one, BECAUSE you aren't married to him.
It's hard to think in these terms, but you need to make sure that you aren't being used as the person who makes this man's life easier with his children. I can understand that he would want to make sure that a future wife would be good with his kids, but you need to know that he is considering you for marriage before you give your heart to these children. Both for your own emotional well-being, and for this man who will need to make a commitment in his own mind that he wants to make a commitment to you.
The best thing to do as far as the 7 year old is concerned is to say to him "This is what your dad said, so that's the way it has to be." You don't need to remind him of why he's in trouble. He knows. And make SURE that the father tells the child IN FRONT OF YOU what the "rule" is, every single time. The last thing you want to do is ask this child what his father's rule is. If he fudges or outright lies to you to get what he wants, you are then in the position of having to call him on his lie. Instead, you put the responsibility for who issued the punishment on that person. You aren't that person. This is part of being the STEP. You're not the parent. Let the father BE the parent.