That the page is sent to your browser as HTML is not a defining attribute and could very well depend on HTTP content negotiation.
Yes, why not? Just because file extensions matter to certain systems doesn't mean they do for others, and nothing about a URL to a file is required to match its DOS/Windows friendly file name.
> GET /<artistname>/<albumname>/<songname>/download HTTP/1.1
> Host: fakemusicstore.example
< HTTP/1.1 200 OK
< Content-Type: audio/mpeg
< Content-Disposition: attachment; filename="<artistname> - <songname>.mp3"
TBH I agree, I personally do my best to ensure the extension in the URL matches the document type on sites I run, but my point was that it's not in any way required and it's actually somewhat common for it to not be the case where the person I was replying to seemed to think it mattered.
The hiding of the index file name in a folder is kind of a quirk, it's automatic behavior that is being taken advantage of to make "nice" URIs, but it's actually hiding useful information.
File extensions, while a DOS/Windows thing, I've found to be an extremely useful convention on unix, and linux, and just about any other system I've used (though I can't remember what we did on VAX box we used to use in the 90s).