I could see changing the ethnicity of various characters. Ethnicity's pretty much interchangeable (as opposed to culture, which isn't, but they aren't causally related, only correlated by the fact that cultures tend to arise around ethnicity--for example, where there are lots of Japanese people, like in Japan, you tend to get a lot of Japanese culture).
But it kind of mystifies me that some folks will argue for switching gender or sexual orientation. These
aren't really interchangeable. Gender is a strong influence on the way the individual relates to the world, and way the world relates to the individual. This is partly due to socially defined gender roles, and we're fortunately evolving away from that I think. However, I submit that it is also partly due to the difference a Y chromosome makes during cognitive and sexual development.
It's not just background. Women and men tend to act differently. Not better or worse, mind (if I
had to choose, I'd say men act badly more overtly, although women can be more cruel

). But differently.
Making a woman who acts "masculine" is no big deal--Ultimate Black Widow is a somewhat well-done version of this, and frankly I suspect a lot of writers wind up simply writing men with boobs instead of women. This is better than the Melvin Udall method of "think[ing] of a man, and tak[ing] away reason and accountability," but it would produce a character just as
atypical of her gender. Atypicality
isn't a bad thing, in fact it's a good thing for creating vivid characters, but when it's treated as the norm, it loses it's specialness.
If there's a demand for specificity about what I mean by differences, here goes. I'd say that men tend to be more impulsive, less caring, less emotive, particularly when it comes to weakness, more ready to force an issue through physical violence or other forms of intimidation, and are often more interested in useless minutiae. Women tend to be more thoughtful, more caring, more emotive--although perhaps less so than traditionally thought--are more interested in using negotiation or manipulation to resolve an issue, and are more global thinkers. I'd also say that women tend more toward sexual exclusivity than men, as an inevitable result of women being the ones who have to pay the price of sexual reproduction. Naturally, no individual can be typed just by their gender, and I'm sure many men are more emotive, less willing to punch me in the face, give less of a crap about the classes of Romulan starships, and are more likely to settle down with me than many women.
Further, and this is something I find hard to articulate with specificity, I notice syntactical differences in the way men and women speak and write. It's no foolproof method, but I can usually get a sense from written material, outside of context clues or explicit statements, as to whether a man or woman wrote it.
Moving onto sexual orientation, this is even less interchangeable than gender. Peter Parker dating Harry Osborne (or Norman for that matter

) is going to be a
completely different dynamic than dating Mary Jane, even if you switched the socioeconomic backgrounds and interests of the two characters. It could even be interesting--but it's sure not going to be identical.