In the Silmarillion (i.e. in what Tolkien eventually published), there was no notion of original sin. This idea was present in earlier drafts, presumably inspired from Judeo-Christian tradition. While the Silmarillion retains a few traces of Biblical inspiration, the idea of collective sin had completely disappeared. I would say that in fact, in the Silmarillion, there is no notion of one's sins being a matter between one and Eru, except in Melkor's case.
At the end of chapter 1 of the Quenta Silmarillion, Ilúvatar presents to the Valar his general intentions with his Children, the Quendi (Elves) and the Atani (Men).
[Ilúvatar] spoke and said: “Behold I love
the Earth, which shall be a mansion for the Quendi and the Atani! But the
Quendi shall be the fairest of all earthly creatures, and they shall have and
shall conceive and bring forth more beauty than all my Children; and they shall
have the greater bliss in this world. But to the Atani I will give a new gift.”
Therefore he willed that the hearts of Men should seek beyond the world and
should find no rest therein; but they should have a virtue to shape their life,
amid the powers and chances of the world, beyond the Music of the Ainur, which
is as fate to all things else; and of their operation everything should be, in
form and deed, completed, and the world fulfilled unto the last and smallest.
But Ilúvatar knew that Men, being set amid the turmoils of the powers of the
world, would stray often, and would not use their gifts in harmony; and he
said: “These too in their time shall find that all that they do redounds at
the end only to the glory of my work.”
Not only does the Silmarillion not mention sin, but the straying of Men was inherent in their being. Even as they later follow Melkor, they are still acting according to Ilúvatar's will (although Melkor isn't).
I don't think the decline in life-span was directly addressed in the Silmarillion. I refer to Zarkanya's essay on the decline of the Númenóreans for an analysis; the decline wasn't abruptly decided as part of the (literally) earth-shaking changes that marked the downfall of Númenor. It should be noted that the kings of Númenor had a markedly longer life than the general population; that long life was apparently at least in part a genetic trait (to use an anachronic term). Quoting the Akallabêth (the tale of the downfall of Númenor in the Silmarillion), because of their ascendance — the line of the half-elven Elros:
But to Elros, who chose to be a king of Men, still a
great span of years was allotted, many times that of the Men of Middle-earth;
and all his line, the kings and lords of the royal house, had long life even
according to the measure of the Númenóreans.
The Dúnedain, of which the Númenóreans (or at least the bulk thereof) were part, did nonetheless have a longer life-span than other Men. Again, from the Akallabêth:
For though the Valar had rewarded the Dúnedain with long life, they
could not take from them the weariness of the world that comes at last
The notes in the chapter on the Kings of Núnemnor of the Unfinished Tales have a longer discussion on the topic; it seems that Tolkien did not fully settle on the exact rules governing relative lifespans.
In summary, the answer to your first question is yes: Men were mortal by purpose. The answer to your second question is less clear, but the shorter lifespans were at least not the direct result of actions by Ilúvatar.