I don't have any children of my own, but I raise (or co-raise) my stepson, my god-daughter, and my nephew. I introduced them to some basic MA, but I didn't force it on them. I let them decide what they wanted to do (including music, skatboarding, etc.) So far, my god-daughter is satisfied with learning basic self defense and playing sports. My nephew loves skateboarding. He stopped doing the MA a while ago, but we just found out that he's a natural wrestler - One day, they were doing wrestling in PE and he pinned most of the guys on the wrestling team. Go figure. I've given the oppertunity to teach MA to my stepson to his father, who is also a martial artist.
As for being strict... I'm VERY strict. I watch what the kids eat, watch on TV, play (video games), etc. I make sure they do their homework, etc. etc. etc. However, I also answer ALL of their questions. My wife and I deal with all of their problems. I guess that because I'm such a kid at heart, I get along with them very well.