I have some games on the desktop version that I would like to edit on multiple devices. How can I get those files to web view without actually publishing those games?

Once you start editing it in the offline browser I don't think there's a way to move it to the web editor. Dropbox or any online storage is your best bet for sharing files between computers.

Having said that, perhaps it's possible to copy your entire game code from the offline editor and then pasting it into code view of the online editor?

The online editor doesn't provide (to my knowledge) a game-wide view of the script. It does show you pieces when you go to edit individual scripts, but not the entire game at once. So you can't do the copy/paste trick, sadly.

When you say "multiple devices", do you mean different sorts of devices (like iPads, etc), or are they all Windows systems? If the latter, you might be better off sharing the source with Dropbox and then just editing it in place on the different PCs with separately installed desktop versions. Dropbox will then mirror it onto all the devices automatically.

Yeah, I use dropbox with no problems.

Another way of doing it is to have Quest offline editor installed on a PC which is always switched on which you can access remotely using any other device using Splashtop or similar. I can access my desktop right now which is 300 miles away and start editing using my iPad if I want. Obviously there's limitations such as bandwidth and resolution differences but editing text in this way works fine. Obviously bandwidth doesn't become an issue if you're doing it on the same network.