Oh, ok. I thought the gist of this all was that you were trying or wanting to duplicate MUD-like capabilities in Quest. If not, then I'll leave it to you to decide what you're actually trying to do.

As far as your questions go:
1) (See my previous answer)
2) The Quest website already has a login system, so there is some integration with Quest into a website already. The Quest code is all open-source, so it's able to be looked at to see how it works. You might be able to replicate on your website what the Quest website is doing (e.g set the right cookies or whatever that the web player looks for) and then have your own login system. You would need your webserver and database back end and then code how you want it all to work. I have not looked at what is open-sourced in the Quest code to know if it exposes the actual website code. I doubt it.
3) This gets, again, into trying to change the existing website integration that Quest has. Quest already supports saving games. You would either have to see how the Quest code is interfacing with the website and try to replicate that so it just "plugs in" or you would have to customise the code to have it plug into whatever architecture you wish to use.
4) That can be implemented entirely outside Quest as some sort of webpage control that lives on the same page as the Quest webplayer instance.
5) Quest games can be converted to run on mobile platforms, but I don't know what's involved. There is a way to convert a Quest game to Javascript code using an experimental tool, but that hasn't been terribly successful either (based on what I've caught through the forum grapevine).
Alex is the creator of Quest. I don't know to what extent he would want to or be able to help create a customized version of the Quest website (with extras like chat being for you to implement), but I imagine he might if the price was right!