As far as I remember, no one ever doubts that. All of the children of the Mad King were legitimate children. That's why Rhaegar was supposed to be the next king.
Also that is the reason why Robert sent assassins to kill Viserys and Daenerys from the moment they escaped to Essos; because he was afraid they might come back to Westeros. Since they have the right claim to the throne, Robert was afraid they might find some supporters and rebel against him.
That is also the reason why Robert had many spies on Daenerys
No, I don't think we have any reason to doubt that Daenerys is the rightful queen to the Iron Throne. The Mad King might have been mad, but he wanted his family to keep the throne. Thus, he must have wanted to have children.
From A Feast for Crows, we know that when Aerys became more and more insane, he started to abuse/rape his wife.
(A Feast for Crows, Chapter 16, Jaime)
In later years, as Aerys's insanity began to set in, he grew sexually
abusive toward Rhaella. He developed a sexual fetish for fire and
death, only sleeping with Rhaella after he had executed someone by
burning. The final time, just before the queen was sent to Dragonstone
to escape the coming siege of King's Landing by rebels during the
Robert's Rebellion, Ser Jaime Lannister of the Kingsguard recalls
listening to Rhaella cry as the king raped her; when Jaime protested
that the Kingsguard were sworn to defend the queen as well, Ser Jon
Darry replied, "We are, but not from him". In the morning, the queen's
maids reported seeing scratches, bruises and bite marks all over her