In The Return of The King, why does Gandalf take Pippin with him when he goes to Minas Tirith?
Or, in a more exact way, why does Sauron think Pippin has the One Ring?
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You partially answer your first question with your second. Sauron, probably through Saruman, guesses a hobbit has his Ring. And as Pippin is the only hobbit he sees through the Palantir, he probably thinks he is the ring bearer. Gandalf also wanted to protect Pippin from himself by separating him from the Palantir. Going cold turkey.
Pippin looked into the Orthanc-stone, a palantir. In that palantir ('a seeing stone', as Gandalf called it in the movie version), he saw Sauron (and Sauron saw him). Sauron must have assumed the hobbit he saw, would also be the one holding the One Ring.
If that (not quite unlikely!) assumption was made, Gandalf may have prefered to keep Pippin well within his sights to keep him safe.