His hair is well-coiffed.

Messy or dishevelled hair, yes. I don't think we really have any way, in English, of saying that a man's hair looks good, except to say 'His hair looks smart.' We also have the expression 'he is well groomed,' and to groom means to brush a horse but this expression, when applied to a person really applies to their appearance as a whole rather than being anything to do with their hair, specifically.

You could say neatly combed. If I heard neatly combed I would picture a young boy whose hair was wet or gelled, neatly parted with comb marks showing in his hair. Like others have said, if you just mean that the hairstyle looks good, you would say something like, "His hair looks great."

Dishevelled is a good opposite, and is used. We would not often say it is well-combed, but there is nothing wrong with it. More likely is "a great head of hair" or just "great hair"

It is correct. But not what we would say 1. His hair looks good 2. His hair is styled nicely 3. His hair is done nicely I am no expert on that subject though ;)