Is there a way to replace Finder altogether?
There's an Apple-documented general overview of the technologies that specifically support kiosk on OS X — 

Technical Note TN2062: Creating Kiosks

— with a section Replacing the Finder. 


  … Is it possible to remove it from the OS, 


I recommend not attempting to remove Finder. 


  including thing like file open dialogs etc? …


Of possible interest, in Apple Support Communities: 


preferring to not show the sidebar in expanded save dialogues