About elves and immortality

I understood this "fading" to mean that their bodies, which are of the corrupted matter of Arda, are incompatible with their spirits (coming from Iluvatar). Eventually this incompability and the corruption of the body would lead to the body "dissolving" or "fading out", leaving the spirit "free" of its body.

Wait, Men are children of Iluvatar too; are they not? Why do they not fade? Is it that they die too soon for the "fading" process to begin?

Also, on the subject of the One Ring giving long life: Did wearing it once give a mortal everlasting immortality, or did the mortal have to keep it with him? If not, why did Gollum continue to live so long after the Ring was lost to him?