I've really only dappled in VASSAL and never played a full game myself or online against anyone.
There is a pretty good community though and they have a
VASSAL Forum where I think you can find a lot of answers to your questions.
1. Can the Vassal modules be used solo?
Yes, but each module is different depending on the game and the designer how it works
2. Are there any special design considerations for doing so?
Each module is different
3. Can a game be saved in-progress and returned to?
Yes I believe so, again each module is different
4. How mobile-user friendly is Vassal (for a user that wants to play on a mobile device)?
Difficult question with a vague answer. People have really wanted VASSAL for all kinds of mobile devices, it is dependent on the Java engine and works best on Windows or Mac. Many people have tried it using touch screens but the GUI hasn't been developed for it. The program is actually a labor of love by a few programmers and not a commercial product. People are hoping at same point it can be used with mobile devices.
I am thinking about this in terms of
A. a user that wants to play both sides of a multi-player game
You should be able to do this
B. something specifically designed for a single player (e.g. a character creation program or a mini-RPG adventure)
That would probably require some Java programming but I really don't know enough about VASSAL to tell you.
C. a way to attract a new generation of Traveller-philes via their mobile devices
Sounds fun