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.
Pippin asks this very question to Gandalf while in Minas Tirith, and Gandalf answers that it is to keep him out of mischief.
Pippin looked ruefully at the small loaf and (he thought) very inadequate pat of butter which was set out for him, beside a cup of thin milk. ‘Why did you bring me here?’ he said.
‘You know quite well,’ said Gandalf. ‘To keep you out of mischief; and if you do not like being here, you can remember that you brought it on yourself.’ Pippin said no more.
The Lord of the Rings - Book V Chapter 4 - "The Siege of Gondor"
Gandalf's remark that Pippin "brought it on himself" is a reference to him looking into the Palantír, which happened right before they set out. The initial reason that Gandalf separated Pippin was to prevent him from being able to steal the Palantír again.
'I think all will be well now,' answered Gandalf. 'He was not held long, and hobbits have an amazing power of recovery. The memory, or the horror of it, will probably fade quickly. Too quickly, perhaps. Will you, Aragorn, take the Orthanc-stone and guard it? It is a dangerous charge.' ... 'But at the least keep this thing secret. You, and all others that stand here! The hobbit, Peregrin, above all should not know where it is bestowed. The evil fit may come on him again. For alas! he has handled it and looked in it, as should never have happened. He ought never to have touched it in Isengard, and there I should have been quicker. But my mind was bent on Saruman, and I did not at once guess the nature of the Stone. Then I was weary, and as I lay pondering it, sleep overcame me. Now I know!'
The Lord of the Rings - Book III Chapter 10 - "The Voice of Saruman"