Physical death occurs, but I don't think there's any real way to end someone's spiritual existence. In the first place, as long as memories of a person exist, there is something binding their "essence" to the hearts of others, and it's arguably possible to resurrect a person based on that alone. Memories and feelings are more essential to the make-up of a person in KH than are physical bodies, as they constitute the basis of identity and personhood, and the body is really just a form through which these human (and universal) elements pass.
A heart only becomes a Heartless when it is consumed by darkness: this can happen in a number of ways, but it isn't typical of the average person's lifespan, and among those afflicted only a rare few have a strong enough "will" that their bodies and souls continue on as autonomous Nobodies. Most of the time, the body of the person fades, and the heart itself can live on theoretically forever as a Heartless in a kind of suspended state until the darkness consuming it is cleansed (for instance, by a Keyblade), at which point the heart will return to Kingdom Hearts, which is where all hearts are thought to have come from in the first place.
As for what happens to the heart at the point of physical death, or after it has been released from darkness, that has always interested me. Personally, I think the heart is recycled through Kingdom Hearts, has the memories of its previous life integrated into the entire mass of the lifetimes it has lived, and is born into a new form with these traces of its old "self" etched into it like tree-rings-- so more as a matter of karmic intention than active recollection. With each new rebirth, the heart's balance of darkness and light might be tilted one way or the other based on the kind of life it lived, and that causes unconscious inclinations to take root and guide each new form in its progression through each new lifetime.
Although it's just an unlikely theory, it'd be a great way to see MX defeated if accessing KH caused him to inadvertently "unlock" the integrated memories of all the lives his heart had ever lived-- we saw what carrying the hearts and the accompanying emotions of just a few lives did to Sora. Imagine taking on all the pain and joy and wrongdoing and heroism of hundreds, thousands, or millions of lifetimes. The cost of becoming one with KH would then be the sacrifice of one's selfhood: it would have no meaning there, where all of time exists in a singular, endless, concurrent cycle of past, present, and future.