For white's next move, should it be recorded as Nge2 or Ne2 as there is only one knight which can legally move to e2? [FEN "rnbq1rk1/ppp2ppp/4pn2/3p4/1bPP4/2NBP3/PP3PPP/R1BQK1NR w KQ - 0 6"]


If white move his king knight to e2 in the next move, should this move be recorded as Nge2 or Ne2 in the algebraic notation? 

After all, the other knight cannot legally move to e2 in this move.
It doesn't matter. The purpose of algebraic notation is to produce an unambiguous record of the game, not a canonical representation that you could compare character by character.

This is not spelled out on the official rules, but I believe it is the inescapable conclusion considering that some parts of the notation are explicitly optional (here I'm talking about the FIDE Laws of Chess); for example:


  [...] each player is free to use the name which is commonly used in his country [...]
  
  A longer form containing the square of departure is acceptable.
  
  [...] an x may be inserted [...]
  
  [...] ‘e.p.’ may be appended.
  
  [...] The last four [capture, check, checkmate, and en passant] are optional.


That said, the rules are clear that disambiguation is only needed when two identical pieces can move to the same square, so the "correct" notation would be Ne2.