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Don't think they're connected but I believe that madness tends to spread up once it pops up. So if your ruler is mad, his kids are likely to be unstable as well, though I'm not sure of it.
 
Yea, I think schizophrenia, crazed etc. do pass on (parents having it lower the kids MTTH to get it). And stressed isn't that hard to get to begin with.

I have a prodigal kinslayer living nearby, does that count? And maybe had a heretic too somewhere in France. No actual insanity in prodigies in my game yet.