Thanks for the detailed information James!
I don't have point-by-point answers for you right now, but I can say that at this point, the indexing system used by Open Data does not (yet) support services using custom projections, and this is causing a cascade of issues that result in many of the oddities you are seeing.
We are actively working on adding support for custom projections (1.0 release), updating the documentation to add this (beta) limitation clear, as well as a means to inform the Open Data admin of the datasets that have been added to the search index for a site and a listing of those which could not be added for whatever reason (i.e. custom projection at this point).
In the near term - can you publish your map in a standard Esri projection? i.e. one that does not use a projection string? It does not have to be web mercator (aka wkid 3857)
Hope that helps!
Dave Bouwman
Javascript Developer
Open Data Team