Since Trunks' existence wouldn't be at stake, he'd have no problem revealing his identity. Assuming Trunks can survive whilst he's gone, they'd probably come back together during the next trip, though Trunks would enter with Vegeta still in an attempt to know his dad better, meaning the plot progresses as it did in the original timeline. The only difference is that Gohan would likely gain MSSJ from his Rosat trip(s) due to having a mindset closer to his father's. I don't see why he wouldn't bring Trunks along as support, seeing as how the Time Machine is big enough for both of them.
As for Super, Gohan being there to train with Trunks would allow them to stop Shin from dying, meaning Zamasu wouldn't come to this Universe and is instead stopped once the Daishinkan or Zen-Oh become aware of him. Even if somehow Shin wasn't saved, the arc wouldn't change much from it's original self. Black only kept Trunks alive so he could get more Zenkais, so he'd probably kill one of the two half-Saiyans due to not being essential. Biggest difference would be that Trunks dies and Gohan fulfills the exact role he did, maybe hooking up with Mai as well.