If you modify chess so that white gets to make one extra move within the first 10 moves, is it possible to show there is a non-losing strategy for white?
Student T gives an explicit answer, but you can also view it from a game-theoretical perspective.
Suppose normal chess is a draw (or a win for black). Then, white can play 1. Ng1-f3 and Nf3-g1, effectively becoming black and the rest of the game is normal, so it is a draw (or a win for white).
Suppose normal chess is a win for white. This will almost certainly involve moving a pawn two steps from its initial position within the first 10 moves, or moving a bishop, queen, rook or king. You can use your extra move to perform this move in two steps, e.g. e2-e3-e4 or Bf1-d3-e2.
In any case, White will not lose the game.
Yes. 1.e4, then move the white Queen out to f3|g4|h5. Prepare for Qxf7+ then Qxe8 or Qxd7+ then Qxe8. Now white wins because the black king is gone.