If I were to add 색 to end of 하얀 in this sentence, would it still be correct? ex: 저 여자가 하얀색 원피스를 입어요.

If 'wear' designates a state and not an action, this sentence is not correct.

"저 여자가 하얀 원피스를 입어요." can be an action only, not a state.

What's the difference between 입었어요 and 입어요?

저 여자는 하얀 원피스를 입고 있어요 is correct

입고 있어요 would be the correct answer in 2 possible situations.

1. we are watching the woman actually trying a white dress right now.
2. we are watching the woman already wearing a white dress.

입어요 can also be the right answer if

1. the woman often wears or enjoys wearing a WHITE dress, not a dress of other colors. (and in this case 저 여자는 would be more precise unless we stress 하얀)
2. THAT woman wears a white dress not the other women.

so either 저 여자가 하얀 원피스를 입어요 or 저 여자가 하얀 원피스를 입고 있어요 can be correct.

One more thing: the more precise answer would be "저기 저" 여자가.... as the translation of the sentence "That woman wears a white dress." will be already 저 여자가...