1. Does "his hair was flowing" mean "he had messy hair"?
No. It means his hair was long and hanging loosely. As well, rather than being messy, the hair is smooth.
2. Does "these luxuriances" mean "his shiny teeth"?
No. It's referring to the 'work of muscles and arteries', his hair and his teeth. All these are his positive/glowing/pleasant/luxuriant features (luxuriances). As opposed to the less positive features that are later described.
3. Does "that seemed almost of the same colour as the dun white sockets in which they were set" mean "his eyes were the same color as the dark skin around them"?
No. 'dun' can mean dull or drab (lacking brightness) or lifeless (in colour)
"His eyes were almost the same colour as dun white sockets" means his eyes were a faded, lifeless white colour. ie. A faded washed-out white colour.
4. Does "straight lips" mean "lips that are not curvy but are composed of two thick parallel lines"?
Almost, but not quite. The lips are not curvy as you suggest, but they are more likely thin or normal, rather than thick. If lips are thick they will almost always be described as thick. Simply saying straight lips almost always implies the lips are normal, but straight. Thin lips are possible, but not thick.