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I would use "pulled" in the context of a dental visit.

Grammar GeekI would use "pulled" in the context of a dental visit.

Is 'out' necessary? In other words, the tooth had been pulled / pulled out. What about 'taken out'? Can the two phrases be used interchangeably?

Well, AMERICAN English, if you say "I need to have a tooth pulled," you don't use the out, and you don't say "taken out." If you want to be more formal about it, you would say "I need to thave a tooth extracted."

Your dentist will refer to the process as the "extraction," I believe. But while he's talking to you about it, he may just say "I advise having that tooth pulled."

In BrE I'd say "pulled out", "taken out" or "extracted". Just "pulled" sounds weird to me, so that's a difference between BrE and AmE.
"Removed" is not wrong, but is less specific. You can have a lung removed.

I agree with Lewis that 'pulled' alone is not BrE, but 'pulled out'. It is interesting to note this difference between BrE and AmE.

The word 'extracted' is used in both BrE and AmE.

Barbara and Lewis, many thanks for your reply to my query.