I was always kind of disappointed that the show created in the plot of the film didn't become an actual kids show starring Robin as Euphegenia Doubtfire. That would have been awesome.
While that would have been awesome, it was also at the height of Robin Williams' career and at the time TV was seen as the death knell for actors. The best you could hope for would be a cartoon with a different voice actor, but you would have been disappointed.
Oh boy. This is all kinda complex and definitions vary, but very generally:
What Daniel Hilliard does in the movie is technically "drag" in the loosest sense of the term, but he's definitely not a "drag queen". That is a very specific type of gender-bending performance art. A queen is a persona that is typically defined by their exaggeration and/or subversion of gender norms.
What Daniel does is merely playing a female role, and that involves cross dressing. It's a job. He's not out to say anything about gender or play with any social conventions. It's just acting.
Wow, the existence of that joke really underscores how fucked American society already was. Gun culture out the ass. I suppose it's only natural that you can be shot pretty much anywhere today.