There's actually a fan theory that none of the current Lannister flock are actually Lannisters. In the books, it's strongly implied that the Targaryens would regularly consort with the wives of their nobles, and Jaime, Cersei and Tyrion are described as having platinum hair, only a few shades darker than the Targaryen white. Add in the mental instabilities, propensity for incest and some remarks Tywin makes to Tyrion about not being able to prove that Tywin isn't the father, and it starts to look like not a one of Tywin Lannister's kids might be his own.
On the one hand, yeah, it's canon that Aerys was infatuated with Joanna (Tywin's wife) and not terribly fond of his own wife/sister Rhaella, and there's a rumour that he had sex with her at some point, before she married Tywin. On the other, Tywin Lannister does not suffer insults gladly, not even from Aerys, and and much less anything having to do with his wife. Iif Tywin had the slightest inkling that Aerys had fathered his children he'd probably have jumped to join Robert's Rebellion at the first chance he got. Instead, he waited around to see who had the upper hand.
Also, not sure of the timeline here, but I think Joanna spent almost all her time at Casterly Rock after her wedding. And both the Queen and Tywin would have an interest to keep her away from Aerys (Joanna was Rhaella's lady-in-waiting before she married Tywin and it's said the Queen tolerated Aerys indiscretions, but not when it concerned one of her ladies). I don't think he'd have the chance.
I suspect his remark to Tyrion was just out of disgust (and convenience), not because he actually thought he was a bastard.
This would also explain why Cersei and Jaime are relatively normal aside from the paranoia (which is also attributable to their environment) and incest, while Joffrey got hit with the insanity stick so bad...an infusion of fresh blood diluted the genetic causes of their mental issues, but going right back to family lovin' compounded them again. It's likely that their other children will start manifesting similar symptoms as they grow older.
Not necessarily! Most of the Targaryens were super inbred, and while they did have their share of crazy assholes that wasn't
all of them. Daeron the Good, Jaehaerys the Conciliator, Aegon the Conqueror, those guys were pretty decent. And that's just kings, if you count general folk you have Aemon the Dragonknight, Rhaegar, Maester Aemon possiby though he was actually one of the least inbred Targaryens so he might not count.
Also, 3:1 odds that we'll never find out if Tommen or Myrcella go crazy or not because they die too soon.
Yes I know Myrcella already died in Game of Thrones, I'm talking about the books