INS Extras

I think it depends on their experiences as heroes but also how old conner is. If the two of them have had too many negative experiences as heroes, they might not engage the hero conversation at all. I don't think clark nor kara would want a super kid getting himself into situations just because he can take a few hits. If conner is older (I'm thinking adolescent/young adult) they would likely be helping him learn his abilities and maybe every so often allow him to shadow them so he sees what it's like to be a hero.

Then there's the other time where the BEYONDER laid eyes on her once and was head over heels turning the world upside down just so she would love him back but since she was rightfully terrified of him (she kinda died because of his shenanigans) he became super depressed and flew off into the universe

At some point you'd think the government or shield would hit up the avengers like "hey man, y'all gotta get the fuck out of new york, this guy can't keep doing this"

For clark, maybe he does odd jobs where he can utilize his strength or maybe he tries sports and he gets pretty far going pro but decides maybe there's better things he could be doing with his abilities so he turns to journalism and since he has these powers he can do things and get to places that the average journalist can't, making him all the more effective. I don't think clark would have any sort of hideout beyond whatever apartment he's living in unless we say he has a secret hideout back in Kansas.

Assuming the public hasn't already seen superheroes I think clark suddenly appearing would shock them because who's this guy flying and punching someone through buildings? Where did he come from? Why is he doing that? Who is he fighting? This would immediately hit the mainstream news and Lois referring to him as Superman is likely how clark gets the idea for his hero name but I think the title of "Superman" may not have the positive connotation that it does (at least not initially).

As for the on/off switch, I don't know if that would particularly affect how they use their powers beyond being able to hide it easier.

In theory they should be able to handle him, but this is Doctor Doom were talking about so marvel is gonna find a way for him to win through some esoteric unknown magical spell that every sorcerer (even strange himself) wasn't aware of

Imagine if the X-Men were dropped into DC comics, they would be national celebrities almost immediately