I think the thought processes are completely dependent on the environment and the cultural aspects. I will use a few distinct examples from comics, myth and literature to illustrate:
- take, on one hand, the wild elves from Elfquest. Their focus is on the "now", and, in most cases, on survival in a hostile world. Immortality is not something that crosses their mind because the only way to mark the time is through repetitive events (seasons) and exceptional events (births, violent deaths, catastrophic events). They are more in tune with the turn of the seasons than with the passing of years. On the other hand, the secluded elves (Blue Mountain elves) are completely different, and almost insensitive to the passage of time - they have recreated a dream world and everything is unchanging, and they *fear* changes, because it reminds them that their microcosm is not the sum of reality.
- take, on the other hand, the sidhe from faerie myths. They might look almost-human, but they are utterly alien. They obey laws - one might even say *world* laws that make no sense to the average human but nonetheless make them immensely powerful, yet vulnerable to people who can find a new take on those laws. They are entities of contracts, and time has no meaning for them - so little that they can't even understand why it would have a meaning for humans.
- And the Tolkien elves meet humans halfway: human enough to share some thought patterns and habits, such as understanding the erosion of time (on the mind if not on the body), valuing life even in its transient form, and acknowledging the value of community and the threat of violence. On the other hand, they stick to their own because they cannot *relate* to the short-lived individuals (with a few, almost always tragic exceptions) and *do not want to* - they are eternal because they are the first-awakened, and closest to the ideal of world that was sung by the Valar, and humans are different - quick to live, quick to forget, quick to die, a constant reminder that the world is *not* eternal. Elves are not mortal, but they don't want to see death as their neighbour nonetheless. And even among elves, the pull of the undying varies - the Noldor feel it most, and the Teleri feel it the least (and the Vanyar never crossed into the east, so have no wish to "return to the womb", so to speak). Elves in Middle Earth are on borrowed time and it weighs on every one of them.
The problem with Shadow World elves is not that they are immortal. It is that we do not know their origin story and how they live it. They might live it in different ways, but for immortal people, the origin is not myth, it is history. Shay people can watch back a century or two back in time and see history, but anything beyond is the stuff of tales and legends, and the foundation myth is fluid - humans adapt to their environment and reshape their origin myth accordingly. Elves, being immortal, have a different take, because the beginning of the Second Era can be a handful of generations ago, if not less. Even if they do not know how they came to be, their origin myth can be preserved and, if it changes, it is for propaganda needs, not because they can easily forget.
I know most people don't really care about such things (it takes time away from slaying monsters and looting corpses, I guess), but, while the Shadow World timeline does a good job of presenting factual events, it does nothing to explain what people actually believe and why, and while it is not an issue for short-lived races, it is for immortal people - you have no clues about how they cope with being essentially "always there".
The only thing you know is that, barring a violent death, they *are*.
But as a GM, I'd rather come up with foundation myths to explain how elves deal with immortality than change the setting to make them mortal - I guess I'm not looking for an easy way out, rather for one that gives me more interesting story ideas.