I watch Family Guy because it's funny. But I know that it's just making fun of things. For instance, just because they made fun of the A-Team doesn't mean I should go and knock a tree down on someone's house just to save a cat. Now, that being said: I am 21, and know the difference between satire/humorous stupidy and being able to tell what's funny and untrue vs. what is actually done in real life. For instance, just because they put down women doesn't mean I agree with it.
Now, at 13...
At 13, that's a bit of different story, and especially at 8 :O
When I was 13, I wouldn't've been allowed to watch this show, nor would my brother (he's 14 now and still not able to watch it) because when you're young you tend to believe everything you see on TV and pretty soon you'll start taking up these values.
Bottom line: I would not, as a parent of a 13/8 year old, allow my kids to watch this show. Once I realized that they're old enough to see the difference between what they (Family Guy) say (e.g., put-down of women) vs. what is societally acceptable and what actually is (women are, in my opinion, some of the most beautfiul creatures on the face of this earth. That being said, some of them are bad apples--and SOME of them turn out to be what family guy says--but MOST of them are beautiful, intelligent, pretty to look at (I'm Asexual-ish), and so on). Basically, whaT I'm saying is: Make sure that you have instilled great values in you children's heads about women, violence, and the like. And keep it ingrained in themm until they're adults. Then, let them watch family guy. (God, I'd be an awful parent, wouldn't I? >.<)
That's as far as I go. As far as managing the meltdowns...I used to have this problem when I was younger and was off playing pretend in my head when I was 13, and constantly being disrupted. That went away. I still have meltdowns, but it's usually over things that I can't control (ie, my savings being used up to pay my parents' "rent" and my health insurance that I can't afford--COBRA--etc.)