No, but Hyde is implied to have lots of sex while in control, often in ways that Jekyll wouldn't. While Jekyll and Hyde are arguably separate characters, they share a body, and Jekyll is often awake and aware while Hyde is in control. The implications of this may make some viewers uncomfortable, but YMMV.
At least, not yet. Reader be warned; this comic is an adaptation of a work where a main character famously kills himself. As of yet, there's no way to tell whether this is how this comic will end.
No genitals are shown, but Hyde tends to joke about sex, and some of the readers' comments and author's notes bluntly discuss who would have sex in what way. Later chapters also feature partial nudity and sexual situations, including a scene that is very clearly the aftermath of two characters having sex, but no one is fully naked or has sex on screen.
Lots of both, especially when it comes to ghosts and hallucinations. At one point, Jekyll hallucinates that another character's face is melting off his skull, and it's shown in fairly graphic detail, just to give an example.
No, but this is Victorian London (or a steampunk version of it, anyway), and it's implied that some of the Victorian views on homosexuality drive a wedge between two male characters who previously had a fling, and there's some internalized homo/queerphobia. Most characters are canonically bisexual or otherwise queer in some way, and the author is a transmasculine queer person, so obviously homophobia and queerphobia in general is portrayed negatively.
Not intentionally, although a character thought to be male is revealed to be a woman. Another character is revealed to be a trans man and is always referred to by the correct pronouns and gender.
Jasper is shown wearing a dress in a flashback, but this is justified by him being trans and having been a young child at the time. If I recall correctly, in the webcomic's comments or in the author's notes, Hyde was once drawn in a dress.
No, but implications of era-typical homophobia/queerphobia, and also a werewolf is hunted by an angry mob and almost carted off to prison for being a werewolf. The society that Jekyll has founded is a refuge for 'mad scientists'; it's implied that science in general is frowned upon.
Very tricky, because Jekyll and Hyde are arguably the origin of this trope, but at no point is Jekyll ever referred to as having DID or being a system. Still, just like with all adaptations of this story, your mileage may vary.