"What colors look good on me?"
You should think of "很" as modifying "好看" here, not just "好".
It's common to use "很" to make an affirmative sentence sound less abrupt and to remove any trace of implied comparison (X is 好看 as compared to an unstated Y), and it doesn't necessarily add much if any emphasis. However, this principle is less applicable where "很" is modifying a two-syllable adjective or adverb, e.g. "好看" versus "好". (The euphony considerations change when the number of syllables changes.)
The same euphony and implied-comparison considerations don't really apply to questions. Here, then, adding "很" sounds a little strange, and somewhat presumptuous: "What colors look very good on me?" (I.e. while the notion of "very" is weak in an affirmative statement using "很", it pops out a bit in an interrogative like this.)
I still think "我穿什么颜色很好看" should be accepted. My reasoning points are these: 1) "好看“ is an adjective. Even though 看 is a verb, 好+看 becomes an adjective. Source: https://www.digmandarin.com/use-hao-in-chinese.html 2) We need 很 to combine Nouns with Adjectives. For example, we cannot say 你好看. We have to instead say, "你很好看", even though we are not trying to say “You are VERY good looking". Source: https://resources.allsetlearning.com/chinese/grammar/Simple_%22noun_%2B_adjective%22_sentences
So, although I feel like omitting the 很 is incorrect, I definitely think that keeping the 很 is not incorrect. Sorry about the confusing double negations in my last sentence :)
Just be aware that the sites that say we have to use "很" often (1) overstate the matter for teaching beginners, (2) leave out considerations of tone (not "tones", but the speaker's attitude as expressed by word choice), (3) leave out considerations about the number of syllables of the adjective, which can change the euphony equation, (4) don't consider interrogatives, and (5) don't consider sentences presenting a choice.
All of the above have bearing on whether or not "很" is an appropriate addition to the sentence, and make the matter more complicated than a simple rule can express. I've provided my sense of the matter from my own experience and learning (including living, studying, and working in Taiwan and China for a number of years), but I'm not a native Chinese speaker and would be happy to hear from any that want to weigh in.
The simple answer is that 很 is not required for adjectives in questions. There are other exercises on the site that show the same.
你弟弟很矮 = Your younger brother is short.
你弟弟矮吗 / 你弟弟矮不矮 = Is your younger brother short?
See the comments by Keith_APP and josslucio: https://forum.duolingo.com/comment/25397648/Is-your-younger-brother-short?comment_id=26600035
I want to put a "得" in the sentence in your first paragraph before "好看" to "correct" it, but I'd like to hear from a native speaker as to whether either version is actually correct.
The suggestion in your second paragraph doesn't seem wrong to me and seems to have the same meaning, but again I'd like to hear from a native speaker.