I'm 99% sure I'm right but I just want to clarify:

My dad is learning Japanese through pimsleur right now because he's interested in Japanese movies.

One thing they mentioned in one of the later lessons was 赤ワイン.

He was surprised to hear that it was pronounced あか　ワイン after I lectured him on the reasining behind putting 'the い' at the end of adjectives when conjugating with a noun.

I was pretty surprised at first too since I'm pretty crappy at grammar I came here.

It's あか　わいん because they translated it as a compound word with two nouns, and the あか is not directly modifying the ワイン right? Do you get what I mean, like it's not saying a "wine that is red" it's literally saying "red wine" sort of like if I said "Stop Sign" or "Cola Drink"

Yeah, that's pretty much right.

This is something you can find in other languages, too. In german, for example, "red wine" is just one word: "Rotwein", whereas in two words it would be "roter Wein" (which means "wine that is red"). So, here, too, the adjective ending "-er" is dropped.

And note that you can also sometimes see some distinctions here; 赤いワイン would just sound like wine that is red rather than the set word "red wine" (although it would probably just be interpreted as a mistake).