I think the best cyberpunk makes a commentary on the anxieties and grotesqueries of our world by drawing an extreme version of it. All the classic cyberpunk is a reflection of the time; the Yellow Peril, the massive increase of corporate power following the Reagan-Thatcher era, and the fears of nuclear annihilation are all reflected pretty well in the original tabletop roleplaying game and Gibson's cyberpunk phase.
If you want to just write about street samurai and netrunners that's totally fine, as long as your characters are stylish and cool and maybe there's an elaborate conspiracy with shady corpos and the CIA.
If you wanted to take it a step further, I'd say you should try and observe the world around you, predict how these current trends would look in a sci-fi dystopia, and put them in your work.
Chrome! At least that was the official answer in the original cyberpunk game.
OK more seriously, the defining feature at least for me is an ecology of subcultures with very different opinions attitude and lifestyles. The future is everything that ever was, just more so.
Even more verbose version: Everybody is into some kind of overdrawn style. So you might be chased by a goon who is into a 1980 style, so he wears an ill-fitting 1980s suite, has an old fashioned pistol and drives what looks like a 1980 police car, only the suit is bulletproof, the pistol fires micro-missiles (but has a little speaker for the authentic bang-sound) and the car can fly. Around the next corner there is a group of punks, they look like 1970s-London but the wear little devices on their ears that project old sex pistols footage onto their mohawks. You run into a hotel that is roman republic themed. Guests are given a complimentary toga. On a screen Julius Cesar gives a speech. Of course it's not a recording of the real cesar, but the studio that produced it rendered it as a black and white silent film with old movie artifacts to make it look authentic (If it's not 3d it must be ancient, right?)
I think a story game needs a story, don't think open world would fly unless you made a huuuuge effort.
Recommended Reading / Viewing
Burning Chrome is Gibson and it fucking owns, as well. It's way more concerned with street scum than Neuromancer, which is more about the machinations of the elite. OP is writing about a street-level criminal, so he might get more benefit out of it. It's also just an entertaining heist story in general.
Quick suggestion from a nobody.
Make the story using the gender and sexual preference you are interested in writing, then if you feel up to a fairly significant amount of work, add those features.
The 21st century thing would be to make it selectable ... and have a lame ill-fitting story as a result. Think of a random story and switch the gender of the protagonist, does it work? I think you could almost get away with changing Frodo's gender in Lord of the Rings, though the interaction with Sam would change quite a bit. Changing the gender of any other member of the fellowship of the ring would change the social dynamics substantially and some subplots would stop working entirely. With stories that delve deeper into relationships the effects would be downright comical.
So either you don't care about the intricacies of social interplay, or you would have to rewrite a substantial part of the story. Taking overhead for scripting into account it would probably be less work to write a different story. And, that is perhaps the best solution. Just make one story that fits for a certain gender and orientation and then perhaps a separate one for something else.
There is absolutely nothing wrong with the protag being a faceless agender mannequin in a cyberpunk setting.