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Your elves will end up cautious.

The only way an elf can die is by accident or violence. All other things being equal: the birth rate amongst the various elf demographics is even and pegged to the species death rate, but the inter-group death rates are not equal, therefore the groups that grow the fastest are the ones that die the slowest.

Therefore: groups of elves whose cultures endorse caution will end up being the largest as their hastier brethren die and get reincarnated into the more cautious group.

This does assume that the rate of elf... erm... procreative events is uniform.

Otherwise: cautious, promiscuous groups grow fastest.

First, even before considering demographics.

The main problem is, with ageless bodies, the pressure to have descendants lessen or may completely disappear. This kind of immortality brings a whole lot of problems and changes, which are partly solved by the limited numbers of souls. Overpopulation for one.

The elf civilization will have two repeating stages. Growing and peak.

The growing stage:

The elf population starts with low numbers. They will have limited and pretty unbalanced civilization, scaled towards the oldest and most knowledgable surviving elves talents and interests. They will make baby elves regularly and without care. The oldest are honored as keepers of knowledge. There may be elves out there who lived through numerous civilizational cycles. Their lifespan will be truly ageless and transcendent from worldly affairs, creating a mini council and sociaty of their own, to guide the "repeaters".

The peak stage:

The elf population reached its maximum. During the growing phase, there may have been conflicts during the families, but they put that behind themselves, by either uniting under a single family (ohh, sooo much incest!!!) or various families coming to agreement. The latter shouldn't be hard as with ageless bodies, much of the biological pressures which causes conflict are mitigated. You can simply wait your turn, improve existing technologies to have more.

The oldest elders are still honored, but there will be a shift. Having a child will be viewed as a gift, a blessing for those who want them. The youngest will be focused on by the whole community and they will be mini celebs, with all the resources at their disposal. Which will spoil them rotten, hastening the collapse of peak stage. Just look up the chinese little princes, with four grandparents. The same here just worst!

Babymaking will be either heavily regulated. It may reach sacred, ritualistic levels with priests and observers as the couple "summon" the spirit repeatedly. (Elf porn ftw!) The right of having one given to the priviliged or those who are next in line. Or they will slowly descend to nothing else but constantly trying to make babies, to increase the chance.

The truly desperate may kill the fellow elves to "free" their spirit... and fornicate on their victim's dying body to capture its spirit (Ghostbusters elf edition?) ... And its just part of the fun. In a max out population, with managable numbers like one million, each death is quickly noticed. A new pregnancy too. Which will lead to "spirit" lineages and other esotheric stuff.

The parents will give birth to the killed elf, good things memories aren't carried over... the dead's family will see the newborn as a lost loved one come back, made worse that relationships may lasts hundreds of years. All kind of insanity and craziness ensue, which will lead to the collapse and mass death of the elves.

The cycle will start anew, with the lead of the wise old who may have many such cycles behind them. Some of those may try to fix the problem, some may be dumb to the whole thing, as they lived through it so, so many times. Suicide among those may not be uncommon or negatively judged...

It would lead to all kind of interesting tradition, worldviews, religion, temperament, technologies and individuals, especially among the truly old.

Elves will value war.

During wartime, childbearing is encouraged, children are valued, and elders are revered.

During peacetime, childbearing is strictly regulated (both socially and with the inability to conceive imposed due to lack of souls), having too many children is seen as selfish, and elders are scrutinized for any sign of faltering (there might be more "accidents" or a ritual ending).

War is messy and inconvenient though. This means the elves will develop social class (or reinforce what they already have). The elite live in well-protected cities and the soldier-class elves fight for honor, land, glory, riches, whatever motivates them.

Just enough to keep the overall elf population down to a manageable level. By which I mean, the elite can have all the children they want and not suffer any social push-back from doing so, or from keeping their elders around forever. Soldier-class families can do the same, though they won't have many elders and the children will learn to fight and go off to war as they reach maturity, so there isn't time to have a lot of kids.

The elite population will increase though and, eventually, might reach a critical mass. No matter. The elites are already stratified and the lowest of them must now send their children (and all future generations) out to be officers in the fight for justi...whatever.

Those children raise their families out in the danger zone and population equilibrium returns. Lather, rinse, repeat.

Such a reality could also create wandering elves. As having families is restricted to those who are lucky/rich enough to win the child lottery, the idea of perpetually living in the same groups could cause some to leave to find new experiences outside the elvish realm.

They might try and form families with other races like humans, potentially with offspring who would then have human souls. You could have a whole other issue if one of these mixed offspring was to get a precious elvish soul - there would likely be some group or certainly families that would seek to kill these mixed people to free their souls for use by the core elvish population.

There might be rituals developed so that someone who is killed can have their soul imparted into a pregnancy straight away. That all depends on how things work. Are souls bound to a body during conception, before birth. Can a female get pregnant then hold the baby at an early stage of development until a soul is ready? Are souls only ever randomly assigned or can the process be manipulated?

Going back to the wanderers, they may also be targets for assassination to free their souls or simply to bring them back to pledge allegiance to one faction or another. In a limited pool, numbers would be important and controlling as many as possible would be key. I don't see any impoverished elves in such a situation as the rich would use their wealth to buy the support of those with less. This could then lead to large numbers of elves becoming lazy as they see their existence is more important than their efforts - conversely lazy elves might be killed to make room for potentially more productive elves.

Lots of possibilities here, interesting question.

As a direct counter to one of the other answers, bear in mind that some of the points below may be mutually exclusive;

Your Elves will end up reckless

Once the Elven population reached ElfMax, each individual knows that their death will be followed very shortly by their rebirth. Sure they'll forget things, but members of a society with such strong evidence of the continuation of the soul may regard death and rebirth as important life stages.

Health Benefit

Even if you're not ageing, that doesn't mean you haven't let yourself go. Accident or over indulgence can cause issue for everyone. Eventually a renewal of your policy is called for, even if it does mean you have to watch the Harry Potter movies again.

Draconian Society Management

Everyone lives for ever, and in an interconnected society? That guy who keeps cutting me up on the tree branch is making my blood boil every god damn day. And he feels that same about you. Eventually the magistrates have to step in and assign both of you to reincarnation to get over it. After all it's still you, and you shouldn't have let this get so far out of hand.

Chasing the Power

The question says that mythical things can carry over to the re-incarnation. Just like the bank accounts in Jupiter Ascending, well every family wants those mythical powers and/or cash for themselves. If the family Assassin can take out the current holder of the AllSpark, just as our team of marathon procreaters are getting started, then we can control that power.

Personal Situation Improvement

Maybe it is just not going well; homeless, broke or unlucky in love? Why not roll the dice again? I could be a Prince next time.

What is this "spirit" that is inherited through reincarnation? It seems to me that a person is very nearly completely defined by his memories, particularly if you extend the concept of memory to include all changes to the brain that result from perception. That is, changes to personality that result from experience, as well as conventional recollections but excluding physical damage to the brain. The personality will also be affected by the body in which it finds itself, but this is pretty well covered by genetics and the accidents of development.
Without memory how could anyone (including the person reincarnated) determine who had been reincarnated where? In any case, what significance would it have?

Presumably there would be some kind of spiritual civil service to make sure that the curses and prophesies are correctly allocated.

On the other hand, if memory was inherited physical evolution could be accelerated, as anyone finding themselves in an unsatisfactory body could kill it and hope to get a better one next time.

As long as curses and prophesies are the only tangible attributes transferred to a new incarnation they will be what determines rational behaviour. As an example, imagine that someone was cursed with death by dragon fire. There would then be a clear incentive for the rest of the population to put him in the path of a belligerent dragon to avoid the risk of someone's child inheriting his fate.

Similarly, if a person of whom it was prophesied that he would bring great wealth and honour to his parents were to die with the prophecy unfulfilled this could prompt a surge of copulation by people hoping to capture the spirit, and hence the prophecy, in their child.

I would expect people who had, or at least could convincingly claim to have, a good understanding of the way in which spirits were allocated to new bodies, and who could advise prospective parents on the most propitious times for conception, to be a prominent feature of the society.

Of course this assumes that curses and prophesies are tangible phenomena. Otherwise all you have is another superstition, of which you could find several similar examples in human culture.

This system would have some interesting consequences with regard to the criminal demographic. To start with, death is not the end of the world, but a (monumental) setback. Therefore, murder can be seen as comparable to theft or vandalism. This also means that the death penalty is a less harsh punishment as it is a free "get out of jail"-card. Even better in fact, since the criminal starts with a clean slate. However, curses are transcendal, so a logical solution would be curses which would mark criminals until their punishment is finished. This would result in a caste of "untouchables", similar to real-life communities who believe in reincarnation.

Considering that elves don't age, and the death penalty is less harsh, the most severe punishment would be livelong imprisonment. This punishment would be reserved for the most monstrous criminals who are not allowed redemption. This system would ensure that they suffer as much as possible, but not being allowed to die, creating a Hell for the living. On long term, this would mean that the most ancient elves are not the wise and respected elders, as one would expect, but the most heinous criminals.