Point of order, it was actually Jon Arryn who started the rebellion. He was the one who raised his banners, rather than give Ned and Robert over to Aerys and certain death. Of course, it was the result of Ned's brother and father being put to death, after his sister disappeared with Rhaegar. But the initial act of rebellion started with Arryn.
It was cold blooded how Jaime and Ned were talking about what happened to his father and brother and Ned said "And you just stood there".
Basically, the entire kingdom just stood there. Everyone knew that Aerys was batshit insane. But he was the king, and as long as he wasn't directing the crazy at them, everyone just stayed quiet. Ironically, Jamie was downright heroic by killing Aerys after he gave the order to blow up Kings Landing. Yet he was still branded Kingslayer and had his honor questioned since then.
She could have put on her "I forgive you for your indiscretion" routine for show and raised Jon as her own son. And probably would have, considering the value she placed on family.
Plausible deniability. She genuinely hated Jon for being Ned's bastard. That meant that there was no question that he was Ned's bastard. Plus, I'm guessing that Ned's promise to Lyanna overruled any devotion to his own wife (who, at that time, he hardly even knew).