I will answer in reverse order:
- Does opened eyes mean that they were initially closed (opened is a verb, and eyes is the object), or it is a participle clause meaning with open(ed) eyes (the eyes were initially open)?
The first one: opened is a verb and eyes is the direct object. The construction with opened eyes is a valid one, but it is not being used here. Incidentally, with opened eyes would differ from with open eyes.
- What is the difference between this phrase and opened his eyes?
His, the other third person singular personal pronouns her and its (and for some speakers, singular their) and the definite article the are determiners. They precede a noun and 95% of the time mark the noun as being definite, as opposed to indefinite. According to me, the lack of a definite determiner here, however, does not necessarily mark the noun as being indefinite. This may seem surprising, but some usages in English show that this is the case: one is list-making. When a native speaker makes a list, he will often leave out all articles. For instance, if I am preparing for a party, I might make a list of things to do, and I will omit the articles, even though I know which things (referents) I am referring to:
1 get haircut (notice: not 'get my haircut')
2 check gas level in car (not: check the gas level in the car)
3 buy gifts (not: buy the gifts)
4 send out invitations (not send out the invitations)
and so on...
I have no conclusive word as to why native speakers make lists in this way. It resembles "headline English", which is another context in which articles and determiners are omitted; I suppose both these contexts could be considered space-saving. A headline has to fit into a certain limited space above the news article; notes can be written on scraps of paper that do not have a lot of space.
There is another usage I can think of, when it is just not important to use the definite article or personal pronoun. In a sentence such as
The man ran into the wall, fell down, stood up, checked fingers and toes, and set off again.
Here we know that the fingers and toes that the man checked were his own, but we don't say that explicitly. Why? I do not know. Grammatically, fingers and toes lack a definite determiner (personal pronoun or the) but it is clear that the sentence means the man's or his fingers and toes, since no other person is mentioned. But even this usage is rare.
- Why is there no definite article or pronoun in front of eyes?
Ultimately you are asking us to guess. We cannot read the writer's mind.
Two users (TRomano and StoneyB) have already given you guesses. I could add another possibility and that is to "depersonalize" or "deemphasize" the eyes, or perhaps the person (aka figure) whose eyes they are. But we cannot conclusively say that the intended meaning is
since according to the Marrowind wiki about the quest Sleepers Awake there are several dreams about this figure that the game-player can have, and two of them do not hesitate to refer to the figure with the masculine pronoun:
I had a disturbing dream. I can only recall one part. A tall figure with a golden mask led me among the dead as through a wedding celebration. I heard many voices, but no lips moved. I strained to breathe, but my chest didn't move. The tall figure spoke with each figure as he passed among them, laughing and joking, as if they were alive, but they made no reply. I tried to cry out, but without breath, my tongue fluttered in vain.
I dreamed that a tall figure with a golden mask spoke to me, but I understood not a word. He smiled, and seemed pleasant, but when he reached to touch me, it terrified me, and I tried to escape, but I couldn't move. I tried to cry out, but I couldn't make a sound. The figure kept smiling and talking, but I felt sure he was trying to cast some sort of spell on me. When I woke, I couldn't recall how the dream ended.
As you precede in the game, or just read the wiki or forums, it is clear that the gender of this figure is male.
But it is just possible that in the version of the dream you ask about the game wanted to keep an air of mystery about the figure. Or it could be just a mistake (unintended omission). What we can definitely say is that the omission of an article is strange and rare here. There can be no conclusive answer to why short of asking the author.