How would she deduce that when all transmissions of the fight got cut off when Gohan went SS2? Only way I could see this happening is if Gohan goes into full detail in his account to her, which is a detail he probably wouldn't want to tell her.
Let's say he does. Gohan would either be adopted by Bulma and live at Capsule Corp., live at Kame House with Roshi and Kuririn's family or go live with Piccolo. I'd say at Capsule Corp. or Kami's Temple would be the most likely options, thus even should Gohan continue his studies he wouldn't neglect his training. In either case, he's strong enough to kill Dabura 7 years later and avoid Boo's revival altogether.
However, whether he lives with Bulma or Piccolo would affect Super's events. Beerus wouldn't destroy Earth either way with him getting the Pudding he wanted, but he'd eventually demand the Super Saiyan God appear nonetheless. If living with Bulma, he'd probably get the chance to be introduced to a girl like Videl to impregnate with Pan or an equivalent and thus allow Super's events to transpire as normal with the possible exception of Zamasu's ambitions never unfolding due to Gohan not having his father's battle lust. If living with Piccolo, however, this would likely never occur and even should Beerus spare the Earth, it would be destroyed along with all the main cast once Freeza makes his 4 month training GAINZ.
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