It’s not a dumb question as such, you’ve just got the timeline messed up, and also the person executed.
Rickard Stark (not Karstark) was executed, burned alive, some time before the Rebellion officially began. (This was at the same time as Brandon Stark’s execution by strangling, and also concurrent with Aerys’s execution of Brandon’s companions and their fathers by unknown means.) After this, it was Aerys’s demand for the heads of Ned Stark and Robert Baratheon that caused Jon Arryn (their guardian) to raise his banners, and officially begin the civil war that became known as Robert’s Rebellion.
The Rebellion lasted about a year or longer. Near the end of it, when it became clear the rebels were winning, Aerys executed his Hand Qarlton Chelsted, who had resigned when he discovered Aerys’s plan to burn down King’s Landing. The execution was performed by Aerys’s new Hand, the pyromancer Wisdom Rossart.
“But Aerys’s new mace-and-dagger Hand was not utterly stupid, and with Rossart, Belis, and Garigus coming and going night and day, he became suspicious. Chelsted, that was his name, Lord Chelsted.” It had come back to him suddenly, with the telling. “I’d thought the man craven, but the day he confronted Aerys he found some courage somewhere. He did all he could to dissuade him. He reasoned, he jested, he threatened, and finally he begged. When that failed he took off his chain of office and flung it down on the floor. Aerys burnt him alive for that, and hung his chain about the neck of Rossart, his favorite pyromancer. The man who had cooked Lord Rickard Stark in his own armor.”
—ASOS, Jaime V
Burning people alive was the only thing that sexually aroused Aerys at this time.
The sight had filled him with disquiet, reminding him of Aerys Targaryen and the way a burning would arouse him. A king has no secrets from his Kingsguard. Relations between Aerys and his queen had been strained during the last years of his reign. They slept apart and did their best to avoid each other during the waking hours. But whenever Aerys gave a man to the flames, Queen Rhaella would have a visitor in the night. The day he burned his mace-and-dagger Hand, Jaime and Jon Darry had stood at guard outside her bedchamber whilst the king took his pleasure. “You’re hurting me,” they had heard Rhaella cry through the oaken door. “You’re hurting me.” In some queer way, that had been worse than Lord Chelsted’s screaming. “We are sworn to protect her as well,” Jaime had finally been driven to say. “We are,” Darry allowed, “but not from him.”
—AFFC, Jaime II
Shortly thereafter (as Rossart was Hand for only two weeks), the Battle of the Trident took place, and the royalist forces were defeated, with Robert Baratheon killing Rhaegar Targaryen. When word reached court, Aerys sent Queen Rhaella and Prince Viserys to Dragonstone for their safety.
Jaime had only seen Rhaella once after that [night with Aerys], the morning of the day she left for Dragonstone. The queen had been cloaked and hooded as she climbed inside the royal wheelhouse that would take her down Aegon’s High Hill to the waiting ship, but he heard her maids whispering after she was gone. They said the queen looked as if some beast had savaged her, clawing at her thighs and chewing on her breasts. A crowned beast, Jaime knew.
—AFFC, Jaime II
Viserys had been a boy of eight when they fled King’s Landing to escape the advancing armies of the Usurper, but Daenerys had been only a quickening in their mother’s womb.
—AGOT, Daenerys I
And shortly after that, Tywin’s forces reached King’s Landing and there was the Sack; Jaime killed Aerys; Elia and her children were murdered on Tywin’s orders; the rebel army also reached KL; and Robert was considered to have won. But the Targaryen forces on Dragonstone continued resisting the “usurper”, and Rhaella and Viserys remained protected. (Note Rhaella crowned Viserys king during this time.) And then:
[Dany] had been born on Dragonstone nine moons after their flight, while a raging summer storm threatened to rip the island fastness apart. They said that storm was terrible. The Targaryen fleet was smashed while it lay at anchor, and huge stone blocks were ripped from the parapets and sent hurtling into the wild waters of the narrow sea. Her mother had died birthing her, and for that her brother Viserys had never forgiven her.
—AGOT, Daenerys I
So, yes, it is unfortunately evident that Dany was conceived through Aerys’s rape of Rhaella after he burned Chelsted. (It is absolutely certain that Dany does not know this, as Rhaella did her best to keep Viserys ignorant of his father’s madness. If he told Dany anything at all, or if she thought about it herself, she most likely imagined her parents had some sort of passionate farewell. Barristan possibly knows the details, but it’s anyone’s guess if he’ll ever tell Dany; I have the depressing feeling it may be Jaime who eventually tells her.)
As for the “child of three” line from the House of the Undying prophecies, I sincerely doubt that the “three” refers to Rhaella, Aerys, and Qarlton Chested. (Bad enough that Dany’s a child of rape, but thinking that the murder that sexually aroused her father would be her “other father”? yeeuuughh.) Since the line is followed by “three heads has the dragon”, I believe “child of three” means that Dany is one of three children who are the heads of the dragon. That is, the line should not be read “child of three people”, but rather “one of three”.
I hope that clears things up for you.