Re: Dwarves with No Spirits
I haven't had the chance to read the Silmarillion (well, beyond the first chapter) or other books beyond The Hobbit / LOTR, but the way I understood it, elves were immortal because they were the physical embodiment of immortal spirits. If they wanted to pass on to Aman (Heaven), they had to actually go there physically, sailing "the Straight Way" across the ocean. If their physical bodies died, their immortality died with them; they were gone, end of story. This is why the elves always considered it such a massive, tragic loss when one of their own died... there was no afterlife for a dead elf, only for the living.
This was also why elves said that humans possessed "the gift of death"; when a human died, their immortal soul was freed from their body and passed on to the afterlife (somewhere beyond Aman?). So while humans lived terribly short lives (compared to elves), their immortal existence in one form or another was guaranteed.
As to how Tolkien perceived how these things worked for dwarves, hobbits and other races, however, I have no idea. Perhaps, since they were not mentioned in Eru's grand scheme (only Elves and Humans had a special place in Eru's creation), maybe they really did not embody immortal spirits or have immortal souls... so they lived short lives like humans but died without an afterlife like elves? Who knows...
Of course, in my own fantasy campaign world, I handle spirits/souls differently... (it's a bit complicated to describe here, but let's just say that all living things have immortal spirits, but what happens to them / where they go after death is dependent on which group they belong to: beings with human-level intelligence, all other thinking beings, and all other living beings... with a few exceptions.)