Is it possible to reorder fields with drag and drop?
To answer just the initial question. "Is it possible to reorder fields with drag and drop?". 

Yes, Settings > Fields > Set up as Matrix > You can then add/delete or reorder there using a drag method within the matrix you set-up. (See image below)


If what you are looking for is more control than this. While having some the same issues learning more about the use of Sections and Channels at http://buildwithcraft.com/docs/sections-and-entries has proven quite useful to me.