Valar, and yes, they certainly had the power, but it’s unclear whether they would ever have actually done so. The War of Wrath certainly fixed Morgoth’s wagon, but the collateral damage was inconceivable (and I use this word knowing it means exactly what I think it means). Gandalf never talks of any hope of rescue from beyond the Sea - but then, of course, it’s not Gandalf’s job to do that, but to motivate the Free Peoples to do what they can with what they have.
Has the company been taken over by Polo?
Did the Ring really let Sauron do anything he couldn’t have done on his own?
Short answer, yes.
The FAQ of the rings will answer your question in far more detail than I can. I think you want section E, which deals with the one ring. In particular, this answer enumerates the powers of the one. Other sections fill in the background.
Yeah, good point. What a piece of crap! 
But, I think they do point out that the elves have been leaving and the kingdoms of Gondor and Rohan are not what they once were, and that they would not likely be able to opposed the newly amassed armies of Mordor under a fully restored Sauron. And thus that plan was rejected. . . but, this may not have been explicitly stated.
Plus it’s tricky to compare Sauron from the Second Age to the one we see in the movies or read about in LOTR, most importantly because while the ring makes Sauron more powerful, without it he is less powerful (than he was before the ring) because a great deal of his original power was used and transferred into the ring.
To go out cruising for chicks?
It also seems to be the case that the Ring wanted to return to Sauron, so the attraction ran both ways.
Destroying them is much less dramatic. Just drop them into any glass of water (of Doom).
My world! My beautiful world…
Matching earrings?
I’ve always seen it as, in the RPG Champions terms, he sold off all his stats and then bought them back through an IIF (so they were higher than they would have been). Total power-gamer move.
Since there was just the One Ring, I always assume it was a PA piercing. 
The emissary referred to the hobbits as ‘spies’. Sauron clearly didn’t know what they’d been trying to do.
Gandalf knew* the mission had not actually completely failed, and the ring had not been found. Maybe Frodo had been killed, but the emissary only mentions one ‘captive’ ‘spy’ implying that at least one of the hobbits had escaped and was still alive, free, and able to destroy the ring.
In the book, Gandalf only rejects the terms after demanding to see the captive:
Sauron is very good at getting information out of captives, that taken with the emissary’s reaction tells him that they probably never had a captive, so either Sauron’s forces killed a hobbit, in which case it’s too late to try and bargain for his release; or it’s all a trick, and Sauron just has some stuff, and no hobbit.
- Unless both hobbits were killed, and someone like a rebel orc or Smeagol currently has the ring, in which case Sauron has no control, and treating with him can’t change anything.
ETA: Y’all missed a big reason for why Sauron needs the ring: if you destroy it he dies.
There’s AFAIK no indication in TLoTR that Sauron still cared about Melkor. He used to be his right hand. But since Melkor had been conveniently disposed of, what not to like in becoming the big honcho himself?
The film Larry Flynt would have made.
Well his work in Numenor where he works in establishing the worship of Melkor in place of Eru implies he never really stopped. Still Melkor’s big thing was in trying to replace Eru or wipe his existence from history and place himself as the prime creator. Sauron isn’t that much different, being worshiped as a god-king in the east.
Gandalf & Co. walk into Theoden’s hall. Grima insults Gandalf. Gandalf insults Grima.
Suddenly, Theoden fires Grima, and starts taking Gandalf’s advice.
I am of the opinion that there may have been a ring at work there.
Me too. May have. But I also wonder why Saruman didn’t seize the ring when he imprisoned Gandalf.