In The Two Towers, when Gandalf was giving Aragorn, Legolas, and Gimli news of what he knew, Glandalf said he knew Frodo made the decision to go to Mordor alone. However, what gandalf didn't know was that Sam probably went with Frodo (as Aragorn, etc. had guessed). Okay, so my point/question is: Gandalf knew Frodo was going alone to Mordor because he put on the Ring on top of Amon Hen, and Gandalf could sense his thoughts (and told him to take off the ring), right? But Gandalf didn't know that Sam went with Frodo because Frodo had taken off the Ring by the time Sam was with him going to Amon Law, so Gandalf couldn't 'read' Frodo's thoughts, right? And Gwaihir (spelled correctly?) only saw the Fellowship being assailed, right?

And finally, when Frodo had the Ring on on the peak of Amon Hen, Frodo was described as seeing Sauron's mind searching the hills around Frodo. So, I ask: Did Sauron 'detect' the presence of the Ring when Frodo put it on, and so then started searching the hills of the Emin Muil, OR, did Sauron already know that the Fellowship was somewhere in that vicinity, and was searching a long while, and Frodo, by putting on the Ring, suddenly became aware of Sauron's continuing search of the Emin Muil area?

If any of my assumptions or facts are wrong, please correct me.