I believe it will be a draw. But I cannot be for certain on that. White may pull one out but I doubt it if there is a best move in every position.
What if both players play the "Best" move possible

Theoretically, if both players play the "Best" move then you would either get a draw, or white would win by having the initial advantage. Realistically however, I think that any given move which may be the "best" against one opponent would not necessarily be the "best" against another because of different playing styles. Remember that you are playing your opponent, you are not playing the board. So, Yes. I think there can be more than one "Best" move for any given position, depending on who's sitting on the other side of the board.

Assuming the initial advantage for white is a tenth or two of a pawn, that this advantage will not be expanded on with best play by both sides, then that small advantage is not enough to constitute a winning advantage.
Let's say hypothetically that there's a "Best" move for each move. Whether the reason's obvious, or it may take a hundred moves to figure it out, but there's a "best" move. If two players always play the best move, who will win?