Hmm interesting points. The existence of the houses of healing suggests that there were illnesses - unless they mean healing from injuries only rather than from illness. Given that elves don't get ill and are more injury resistant than men , I always wondered how Elrond got quite so much skill in healing
... but of course he had a lot of non-elf visitors.
I suspect that, in general illness, is linked partially to moral/spiritual decline and that is why in those societies which lead simple, healthy lives, illness is rare. I think somewhere it says that the Numenoreans were less afflicted by illness thatn lesser men but I will check this, Illness in LOTR often emanates from the enemy directly - eg the black breath, or indirectly. Finduilas who is one of the few to die young but not from violence, was "filled with horror" by the shadow of the east. Gilraen also dies it seems from despair in the face of the Shadow.
Will do some research...