yes ArcGIS Explorer is totally customizable. With use of both the SDK and the Application Configuration Manager, you can create the functionality you want, and control the look and feel of the application.
You won't be able to directly drag feature class items into the routing control itself. A Note (point on the map that has a reference within the contents control) can be dragged and dropped into the routing control for use. Yes, request a route, and the result can be saved and looked at etc when disconnected. Simply save the map.
Yes ArcGIS Explorer can be used when disconnected. The imagery will be cached to the local machine for areas visited whilst connected, with the exception of Bing maps, Bing tiles are deleted on application close. So travel and zoom to the extents that you wish to view when disconnected when in the office, you will have them when in the field. Reading on disk feature classes when disconnected is in no way different to being connected. Your developer will be able to query the feature classes for attributes that he can use is forms.
Yes you can control feature class visibility on application start. Set up the map as you want it, with layer on or off, transparency etc, save the map. Next time you open the map doc those settings are honored.
Advice: Play with the application and get a feel for it, read our blog where Bernard covers a lot of what can be done, and what others are doing. Read the help and look to the sdk samples for ideas as to getting started, and some of the more obvious examples of how to do things. Lastly, use the forum as you are doing, great place to get help and advice.
All the best,