How can I share a savegame for Dwarf Fortress between multiple computers (laptop and desktop)? Can one instance of Dwarf Fortress run graphics and the other not?
Yes. When you "Save and Quit" the game is saved to the hard drive. This can be uploaded to other computers. DwarfFortress.exe uses the folder path specified in its config file to look for saves on your computer. It's worth noting that you do need identical "world" files as well or the game will crash.
The easiest way I've found to do this is to use a software versioning system (such as SVN or GIT) and version the entire folder. This will have the added benefit of indexing your saves and allowing you to go back to previous save points.
Your graphics set does not need to be the same. In fact, on a single computer you do not have to have a consistent graphics set between saves. You do, however, need to make sure your config file correctly identifies the graphics set you plan to use.
I use Dropbox! The easiest way is just to drop the entire Dwarf Fortress folder into the Dropbox folder. This would also make it easy to share with others, since you can make Dropbox folders publicly or small-group accessible. (Of course this doesn't allow you to have different configurations, but you might be able to modify my setup for your needs.)