Other answers have made good points about how to react when opponents make unexpected moves in the opening, but I wanted to specifically address one part of your question because it is a common issue, but has been internalized so much by experienced players that they forget that it isn't obvious.
Chess openings are defined by both players' moves, not just your own. The Danish Gambit, as you say, is defined by the moves 1.e4 e5 2.d4 exd4 3.c3, not just by the moves 1.e4, 2.d4, and 3.c3. If Black plays 1...e6, for example, there is no way to reach the Danish Gambit.
To answer your specific questions:
But what happens if the opponent deviates from this move order?
Then you have to find the best move to deal with the different position on the board.
Would it still be the Danish gambit?
Can I still follow the moves in the Danish gambit or do I have to use other moves?
It's possible that any given chess move could be useful in another position (for example, 2.d4 is a fine move after 1.e4 e6 just as it is after 1.e4 e5), but you have to evaluate based on its merits in the actual position. Blindly following a sequence of your moves and ignoring what your opponent does is a recipe for disaster.
Does an opening variation involve any alternative moves?
Sure. For example, the Danish Gambit is defined by 1.e4 e5 2.d4 exd4 3.c3. Black might decline to take the pawn and play something like 3...d5; it is still called the Danish Gambit, but the gambit has been declined. Or after Black does take the pawn with 3...dxc3, White could take it with 4.Nxc3 or offer a second pawn with 4.Bc4; these are both different branches of the Danish Gambit.