How Much Wiggle Room Does A Character Have?

I had a whole other post going, but I didn't like how it was coming together (or wasn't, more accurately). But this came up as I was working through it, so let's go with it instead.

It's in regards to what we'll accept as the range for a given character. It's not about their core characterization, more what you think is essential to their setting. Who is around them, where are they active, what are they doing, stuff like that. It's not what you'd want to see in a perfect world either, but more sketching out the boundaries of what feels right for the character, however you define that. Think about it in terms of directions they took the character you thought worked or didn't.

For example, with Spider-Man, I don't need Peter Parker and Mary Jane to be married, or even dating. They can be, but not required. Aunt May doesn't have to know Peter is Spider-Man, but she can. He can be high schooler, college student, full-time working adult. Peter doesn't have to work at the Daily Bugle, but him as a knockoff Tony Stark is a no-go. I can sort of roll with Ben Reilly as Spider-Man, but not Otto Octavius hijacking Peter's body and calling himself Spider-Man. He can be an Avenger, I guess, although keep their involvement in his solo title to a minimum.

I don't want to specify characters, because I'm curious which characters people would have strong feelings about. But along these lines, do minor characters have a wider range than the more popular, first-tier characters, a smaller range, or does it depend on who we're talking about (and who's doing the talking)? The Shroud has had relatively few appearances compared to Iron Man. Would that restrict what feels like an appropriate setting for him, compared to Iron Man, because we've seen him in a more limited range? Or does it open things up more, because there are fewer pieces that might be considered "essential"? He's fought Dr. Doom, pretended to be a crime lord, made a team out of old Spider-Woman villains, dated the second Spider-Woman, been crazy and fighting Daredevil, been a Hero for Hire. Is there a cast member you think would have to be there, a locale, an enemy, a goal, or is it all open for him?

I feel as though it would depend on the character for me. I think I'm finding I might have a somewhat limited range for Cassandra Cain (or maybe it's the Rebirth version feels like an entirely different character DC is trying to sell to me as being the old one that I liked), whereas I don't have many notions about what I wouldn't accept for Cave Carson. I would expect subterranean adventures, but I'm not going to demand he be friends with Wild Dog and Will Magnus in the future simply because his current series set things up that way. Those don't seem like essential features to his story. But I probably have fewer restrictions on Wolverine than I do on The Ray, so it varies.