when opponent strays from line

According to the chess.com database, that move has been played 20 times in Master games (and the position has been reached 26 times).  The White replies (with the White win/draw/loss percentages) are 4.dxc5 (16 times, 64.3/14.3/21.4), 4.Be2 (3 times, 66.7/33.3/0), 4.d5 (3 times, 33.3/33.3/33.3), 4.Be3 (2 times, 50/0/50), 4.Bb5 (1 time, 1-0), 4.Bc4 (1 time, 0-1)

That happens in every game at some point: one of the players plays something you didn't know or remember from "theory".

Then you play chess.

There is no such thing as a "non-existant" opening. Perhaps this variation didn't get its own name in the particular list you're looking at, or it wasn't even mentioned. Perhaps it was never played before. Doesn't mean it doesn't "exist".

And why "should have been cxd4"? That's just what usually happens, nothing more.

My club experience with this has shown that the one who varies from a known opening on moves 2-4 is either making an inferior move (intentionally or not) or trying to transpose into an opening they prefer.  Usually there is a reason an opening is non-existant, and that type can often be exploited on principle.  Your move 4. c3 looks playable, from an expert's perspective.

This looks similar to the Opera line in the Philidor. When your opponent plays a sideline or a non-theoretical move, don't assume that it is inferior. Play carefully, and try to find the best moves over the board. If you get cocky, you will lose.