I'm basically just trying to avoid hitting G, Shift+Z, Shift+Z every time I want to move a vertex freely on a surface. Is there a way to set this to a single hotkey?
Not sure exactly what that key combination is supposed to do nor if it is supposed to be used in Edit Mode or Object Mode specifically, but go to the User Preferences > Input look for the relevant context under the 3D View category, like Global, Object Mode, or Mesh etc.
Either add a new entr there using the Add New button at the bottom, or modify an existing.
Pick the relevant operator, I am guessing
transform.translate, expand the settings bellow and adjust the available controls to your liking.
I supposed the Constraint and Align with point Normal are of particular interest to you.
Update from a newbie: In Blender 2.80 the dialog is in Preferences > Keymap > 3D View > 3D View(Global) > Move* *note that are three "Move" functions. Edit that with the "G" key selected by default.
Although I'm searching a way to also move objects perpendicular to the viewport, this is useful for me.