Really nice description, and the care and intricacy that is going into your design is really inspiring. I hope we can find a way to bottle what you've learned about trying to make a flexible and obvious UI for your game and pass it on to other authors. Seriously, I wonder if we can put together design diaries and a guide or tips based on your and a few others (Charles, Terri, I'm thinking of you especially, but I know there are those I'm forgetting at the moment too) and make that available to other authors. I think it would help them understand ARIS at a level beyond the basics in a way that the documentation and tutorials never could.
I'll try to help with your difficulties. I think in general your hope to set scene from the quest button as a way to initialize the map for users should be workable.
Focus of exitToTab
Your experience of sometimes sending a player to a full tab and sometimes to one that is pushed off to the side with the nav still visible sounds like some inconsistent stuff inside the ARIS app, a rough edge that hasn't stuck out enough to need polished before. I think the idea would be that you're always sent to the full tab, but I might be missing something.
The code I remember using is slightly different than what's listed in the ARISjs guide, but what's there matches what you found: ARIS.exitToTab('Map'). That is the single quote, the lowercase of the usual double quote. I never can remember this stuff and usually have to try the variations you brought up to test before proceeding. I think you could check for the exact names in the code for the ARIS app, but if you find them, you could also put them into the manual (I would very much welcome contributions from you or other authors to the manual. It's just a google site so it would be easy to do). I would test this somewhere detached from other things that might go wrong to make debugging easier.
I do not know for sure, but I believe there would be no problem with running multiple lines of code within a call.