Is it possible to convert .quest files to an html5 or a .zip with an index.html file or similar in order to share quest projects outside of the quest editor or this website, similarly uploading them to other dedicated game sharing sites like itch.io?
While I was looking around for this I came across a type of converter on github but it said it was outdated and no longer getting updates, so I assume that’s not an internal feature. So are there any clever users out here that's figured out other ways to do it?

The Quest engine runs on the server side. So there's 5 ways to run a game:

1. Use this website
2. Install the server-side component on your own server
3. Distribute it with the desktop player (which is actually the server with an embedded browser)
4. Write a compiler to translate Quest code into some other language (Javascript?)

There are several projects that attempted this, but none was completed. I suggested how one of the biggest issues with it could be fixed, but couldn't implement the changes myself because they were writing the compiler itself in a language I'm not familiar with.
More details on this option can be found in the "QuestKit" subforum

5. Rewrite the Quest engine itself from scratch in another language

I think this should be easy — but very time consuming — in Javascript. The syntax of Quest's scripts is simple aside from a few idiosyncratic features, and the client side part of quest already uses jQuery, so we've got XML parsing built in.
I'm going to fix this some day; it's on a list behind fixing the save system to work for incremental releases, and tidying up the code for the sidebar. So, it'll happen some day, if I can ever find enough time.

The spiritual successor to QuestKit and QuestJS is, I believe, called 'Quest 6'. This is a Javascript project, so will be able to run completely in the browser. However, last I heard it isn't going to be backwards compatible, so won't be any use for running your existing .quest files. You'll likely have to rewrite the game for the new engine.