Bilbo didn't alert Sauron when putting on the Ring because Sauron was still weak and not actively searching, the Ring was distant from Mordor, and Bilbo's use was brief and unaware, unlike Frodo's who was actively hunted and closer to Sauron's power, making the Ring "shout" when Frodo wore it, while Bilbo's use was a quiet whisper. Sauron needed torture (Gollum) and proximity (Frodo in Bree) to pinpoint the Ring's location, not just its temporary use.
Sauron's physical form was just like clothing, really, and he used it as a tool or a weapon. Actual Sauron was essentially formless, so whatever he would have appeared as would have only been what he wanted someone to see—though only as a figure of malice by the time of LOTR.
Sauron, being a Maia, already exists in both, and as such does not disappear when he wears the Ring.
Why was Sauron afraid of Aragorn potentially possessing the Ring? Gandalf explains it in the Return of the King (during the council after the battle of the Pelennor fields) that Sauron would never think somebody would want to destroy the One Ring.
Gandalf pretty clearly told Frodo that Bilbo was very resistant to the corruption of the ring, because he had actually risked his own life to avoid killing Gollum when he took it. That made Bilbo the first ring bearer to acquire it without killing anyone.
Tolkien's description of Gollum conforms to a Catholic – and Thomistic – account of envy, which is a “sadness of the soul”; and it is Gollum's unbearable sadness and his unquenchable desire for the one Ring that marks his character.
Power was never going to succeed in defeating evil. Gandalf was at exactly 12.35% of his full power as Gandalf the Grey, this went up of 18.237% as Gandalf the White. The Maia varied greatly, as did the Valar. Even at his fullest, Olorin/Gandalf was not the equal of Mairon/Sauron, not even close.
He did not trust Gollum, for he divined something indomitable in him, which could not be overcome, even by the Shadow of Fear, except by destroying him.
As a Maia, Gandalf was an angelic being in human form, in service to the Creator (Eru Ilúvatar) and the Creator's 'Secret Fire'. He took on the specific form of an old man as a sign of his humility. His role was to advise but never to attempt to match Sauron's strength.
Elrond chose the fate of the Elves and therefore the choice was presented to his children. The reason why Eldarion and his sisters (the children of Aragorn and Arwen) were born mortal was because Arwen chose mortality and therefore that fate was passed on to the rest of their line.
Sauron needed Gandalf out of the picture permanently, as he was the only enemy with the knowledge, leadership and determination to thwart him. His biggest mistake was ego. He succeeded by trickery, by his ability to deceive others. He conquered so much land that his ability to control became out of reach .
His task was to do the bidding of his masters and so when he proved unfaithful in doing that task his masters stripped him of his robes and gave them to one who would do their bidding. Gandalf is now the White, Saruman as he should have been.
Instead, we have to consider that Sauron, the original owner, is not human, but has a quasi-divine status. Therefore ownership of the Ring is subject to Divine Law. With this precedent in mind, many of the ambiguities clear up nicely. After numerous transfers by violence and trickery, Frodo is the true owner.
It's possible that Gollum's prolonged search for the ring allowed it to maintain a significant power over him, suspending his aging process. In contrast, Bilbo's exposure to the ring's influence waned after he willingly (mostly) relinquished it, allowing his aging to resume more noticeably.
In the books, Legolas actually screamed in terror when he realized that a Balrog was coming. It was the Balrogs that rounded up the elves for Morgoth, to be corrupted into orcs.
But that's actually the answer: Eru Said So . The Undying Lands are a place for immortal beings (Eldar), and mortals are forbidden from entering.
It's not the years; it's the mileage. Gandalf isn't a Maia in the same way Sauron is. He was sent to middle earth as an old man, so his body is much more real than Sauron's “raiment”. He can't use his power to the same extent, he can't change his forms at will.
Sauron's rise to power in the Second Age is portrayed in the Amazon prequel series The Lord of the Rings: The Rings of Power. He first appears disguised as the non-canonical human character Halbrand, and then in the second season as Annatar (a canonical alias of Sauron), both played by Charlie Vickers.
A group of UCL medical students, led by Dr. Liz Sampson, concludes that Gollum was actually suffering from schizoid personality disorder.
Sauron's evil grew from his original servitude to Morgoth, the world's first Dark Lord, into his own obsession for "mastering" all things in Middle-earth through wicked means. After Morgoth's fall, Sauron was fully autonomous and committed, for two Ages, to gaining tyrannical control over Middle-earth.
Gandalf is not permitted to use his full power during the 3rd Age (some exception apply). He has been sent to guide humans to the 4th age, the age of mortals. He can perform minor magic as needed, but he cannot "bust out."
Sauron was originally a Maia of Aulë the Smith; his name then was Mairon, meaning "the admirable." He learned much from Aulë in the ways of forging and crafting, becoming a great craftsman, and mighty in the lore of Aulë's people.
Gandalf was one of the Ainur, specifically a Maiar or immortal angelic being created before the world by Eru Iluvatar (God). He was described in the Silmarillion as the wisest of his race, but not as powerful as any of the Valar (which included Melkor/Morgoth) or Sauron, a stronger Maia.