Ok.
I made a new mmpk, but unfortunately my home internet is too slow to upload it to AGOL so I will have to wait until tomorrow to try it out when I can go into town and get real internet (I live in the mountains).
I believe that the map should be in WGS 1984 Web Mercator. The regular online map used in the app is created in AGOL so it should automatically be in Web Mercator right? I checked the tile projection and it was the WGS 1984 Web Mercator before I made the mmpk. I did have the "Enable Anonymous Use" option and clicked on it when creating my new mmpk.
Just to make sure I did not mess anything up with my "edited" app, I opened a new QuickReport template from scratch, inserted my old mmpk, and there was still no map displayed when I went offline. I have no idea with that file. One thing that that I was thinking is that maybe the offline map does not work because I do not have a developer's account yet? I am fishing for an answer here. Maybe my new one will work tomorrow.
With the new Map Viewer, can users create a new feature and attach an image to it? That is the functionality that is most useful to me in the QuickReport template.