In the land of illusions is death an ending? Or is it a means for a new beginning? If that's not an illusion what is?
Depends on what you mean with "ending" and "beginning".
For the regular humans, and by that I mean the not-Akyuu ones, there are various things that should be held in consideration : if a human dies while bearing some super strong feelings of attachment or regret to the world, he'll become a ghost ; and if he was consumed by grudge, like jealousy or something that goes along these lines, chances are that he becomes a vengeful spirit or a youkai. In these two cases, you can say that the former human still keeps some traits of his individuality back when he was still a human : In these two cases, you can say he still continues on, even after death, so I'm am not sure if you could consider this as a new beginning.
When a more regular human dies a completely regular death, with no problems, no regrets, nothing whatsoever, his soul eventually crosses the Sanzu river, gets to Higan and get judged and, somewhere down the line, eventually reincarnates. Now then, here's the thing about the not-Akyuu reincarnated souls : they don't carry any memory whatsoever of who they were in their previous life. Even the vengeful spirits in imprisoned underground, who are supposed to stay there until they have repented on their past sins, have no memory whatsoever of the sins they are supposed to repent for (a certain character said that, but I don't quite remember). In this case, it's a complete new beginning for the reincarnated not-Akyuu soul : no trace whatsoever of who it was in the past. I guess this might fit what you mean by "new beginning".
In the case of non humans (ie : beasts,...) , supernatural beings and "special cases", I won't delve into the specifics, but basically what should be considered is whether they can die or not and if they do, what happens next : do they stay dead or not ? No seriously, you can stretch this topic far and wide, especially if you take each and every race and peculiar characters into consideration.
All in all, for the humans, it's mainly about their soul : it simply follow the cycle of reincarnation. In other word, it's getting recycled. I am not sure if recycled material can be considered as "continuation" of what they were before, though.
Also if your emotions die out and is replaced with apathy what happens?
You become apathethic. Duh :V
Or you become a background character for some fighting game stage