Hi,
RE: Problem 1:
It is not possible to edit Shapefiles directly via the ArcGIS Runtime SDK for WPF. Instead you will need to move to using the File Geodatabase (which is what happens when you include your data within the Map Package). I am not sure why the EditorWidget is not saving features in your case, are you starting the LocalFeatureService as editable?
RE: Problem 2:
I cannot immediately say why the layer might not be displaying, it could be a problem starting the local service from the MPK (you should handle the service startup completed event, or await the StartAsync Task if you are using async Tasks). There is an error property on the service you can check. The LocalServer also write information to log file, which you can enable and access via the LocalServerUtility app available as a shortcut on your Start screen/menu. It is also possible it is simply a projection issue, perhaps your TIFF and/or the Shapefile and/or the map document are in different coordinate systems.
It is also worth noting, we have been working on a new .NET SDK for the ArcGIS Runtime which includes a WPF API for building Windows Desktop apps. It's an evolution of the existing ArcGIS Runtime SDK for WPF. You can find information on the current beta here: ArcGIS SDK for .Net | ArcGIS for Developers. Rather than using Shapefiles, Map Packages and LocalServices, in the new .NET SDK you can work with runtime geodatabases directly. These runtime geodatabases need to have been created from ArcMap (similar process to the Map packaging) or extracted from a feature service (on premises or hosted online).
Cheers
Mike