I would like to be able to move features by small (preferably predefined) increments using the keyboard arrow keys? Can this be done?
There is no way of doing it with the arrow keys.
But you can do precise movement by specifying a delta offset as Mintx suggests. The one problem with the delta movement is that it leaves your feature's origin un changed and that might be undesireable.
However, a left-of-field quick-and-dirty solution that would not require any programming could be to create a fine mesh over your data (you can get ESRI to do it with 'Create Fishnet' or Hawth's Tools) and then use snapping. Obviously this is VERY 'Heath Robinson' and would be unworkable for huge areas but could be serviceable.