The elves were sailing to Valinor, the undying lands where the Valar (angels) live. Only elves were allowed to go to Valinor.
A special permission was given for Bilbo and Frodo (and, some say, Sam) because they had been ringbearers. Frodo in particular was physically and spiritually maimed from the quest to destroy the Ring.
He was suffering very much because of this, especially on the anniversaries of the destruction of the ring and of the attack at Weathertop, where Frodo was stabbed by a morgul-blade and came very close to becoming a wraith.
It was believed that Frodo would be able to find peace and healing in the environment of the undying lands, which is probably as close to paradise as anywhere you'd find in the world. It was his reward for his sacrifice.
A number of good reasons, the main one being that it's supposed to be a secret quest which under no circumstances could Sauron be allowed to learn about.
There is an oath sworn that only the members of the fellowship should be informed about the One Ring being carried out into the world to be destroyed. It was seen as too dangerous to be openly revealing it, with spies and informants all around.
It's also debatable if the eagles would even be willing to do it, even if they did know. They're proud and holy servants of Manwë, not taxicabs. And if the eagle should fall victim to the Ringlust, then a catastrophe is quite certain.
It's hard to say, I find different parts of different eras interesting so it's difficult for me to rank eras like that. I guess the dawn era, I enjoy reading the different interpretations and myths and how they're similar and different.
Re the Dwemer, I think that their disappearing is their greatest mystery. Some say everyone was taken to another reality, or to a plane of Oblivion. However, unless I'm remembering wrong the Dwemer left behind piles of ash where they were standing. In other words, they all set on fire and burned away.
They did have some kind of telepathic power, so they were mentally linked. I think they operated kind of like a semi-hivemind, and it was this mental link that spread disaster when Kagrenac activated the Heart. Thats why that one Dwemer survived - he was in oblivion at the time and was not in contact with the hive.
I don't think Tolkien would have liked Skyrim. One criticism he had of The Lion, The Witch and the Wardrobe was that there were too many conflicting elements that clashed and detracted from the whole. I think a sandbox experience, and especially a streamlined experience like Skyrim, would not appeal.
Additionally, I think he would find TES lore far too nihilistic for his tastes. He had a passion for writing fantasy that reflected his own moral and spiritual beliefs. I think some of the esoteric concepts like the Tower or Godhead-Dreaming too detached from the story that Skyrim told and which he himself enjoyed telling. Heroics, sacrifice, redemption - these were the themes that Tolkien enjoyed to see and TES doesn't deliver much on that.