As you mentioned Pix4Dcapture can be installed only on Android and iOS devices for now. Using desktop devices with Pix4Dcapture is not in the pipeline at the moment.
It is not possible to directly design and fly a polygonal mission with the Android app at the moment. However, a workaround would be to fly several smaller rectangular missions, to adjust them according some reference points on the map background, and to process all the images together in Pix4Dmapper. This is facilitated since the grid missions are displayed and so can be adjusted on the same map background.
Resuming a mission is currently an option only in the Android version of Pix4Dcapture. In Android when a mission is interrupted because of low battery or bad flying conditions, the incomplete portion is saved so that it can be flown later (e.g. after changing the battery).
Mission resume is not supported yet by iOS version of the app, but it is in our pipeline and our developers are working on it. However, we are not able to give a precise time frame for the introduction of mission resume in iOS version at the moment.
Since the latest Android version 3.2.1 it is possible to save offline background (roadmap or satellite) per project for several zoom levels. They are loaded only if there is no Internet connection. Additionally, in the coming version the option of having alternate map providers will be available. Thus, you will have the option to select an alternate basemap from preset defaults.
Regarding the background map on iOS, it is possible to cache them before going to the field when you have Internet connection in the office. They will remain on the device for 2-3 days depending on how the temporary memory of the device is used. You can take advantage of these temporary background maps to plan flights without Internet connection.