"The milk that the man is drinking is cold."
Translation:남자가 마시는 우유는 차갑습니다.
Yes the word order is dictating everything here. We can make this a basic sentence in English by getting rid of the specifics: "The milk is cold." I'm sure you know by know that's 우유는 차갑습니다. The trick is specifying which milk is cold. In English, all that info comes after "milk" but in Korean it works more like an adjective and comes before the noun. That's why man has a subject particle, because it's in a different clause. In the clause describing "milk," man is a subject, but overall he is not.
남자가 마시는 우유 = the milk the man is drinking
아기가 먹는 야채 = the veggies that the baby is eating
자동차가 있는 길 = The road that the car is on
검은색 것 옆에 있는 흰색 것은 = the white one that is next to the black one
After any of these clauses, the real sentence continues. 아기가 먹는 야채는 맛있어요 = The veggies that the baby is eating are delicious.