Gollum had been suspecting something since at least as early as his conversation with himself in The Passage of the Marshes:
'Yes, yes, my precious,' came the answer, 'we promised: to save our Precious, not to let Him have it – never. But it's going to Him yes, nearer every step. What's the hobbit going to do with it, we wonders, yes we wonders.'
It's not beyond the bounds of possibility that Gollum had overheard something along the way: he had been trailing the Fellowship since Moria, and in The Great River Frodo speculates that he had picked up their trail again when they left Lorien at the Silverlode, which was immediately before Galadriel's gift-giving and Celeborn's discussion of the further routes (to Mordor and Minas Tirith) to take.
The company knew that he was following them by the time they were on Anduin, and it seems reasonably certain that he may even have witnessed Frodo's confrontation with Boromir. In other words, he had ample opportunity to learn of their plans for the Ring, and even if Tolkien doesn't explicitly state when it happened (the movie is more specific: Frodo tells Gollum) it doesn't seem that he needs to.
For the second part of your question, it seems that Gollum's long possession of the Ring had given him some form of attunement to it, as Frodo indicates in The Taming of Smeagol:
Swear by it, if you will. For you know where it is. Yes, you know, Smeagol. It is before you.
Even aside from that, Gollum must surely have been able to locate Frodo's hand (and the finger with the Ring on it) by touch during their wrestle at the Crack of Doom.
For the third part of your question, Frodo's finger was still in the Ring when Gollum fell, as the scene in Mount Doom confirms:
But Gollum, dancing like a mad thing, held aloft the ring, a finger still thrust within its circle. It shone now as if verily it was wrought of living fire.
'Precious, precious, precious!' Gollum cried. 'My Precious! O my Precious!' And with that, even as his eyes were lifted up to gloat on his prize, he stepped too far, toppled, wavered for a moment on the brink, and then with a shriek he fell.