One example of a T Tauri star is the T Tauri system, after which
the star types were name for.
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No, the T Tauri star is not a giant star, in fact it is a
medium-sized star. It is also a very cold star.
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T-Tauri phasee
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A T Tauri star is sort of a precursor of a star, or about to
become a "real" star. Therefore, it can last anywhere from a few
million to several trillion years before it runs out of energy,
depending on its mass.
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No, all T Tauri stars are very young, only a million or so years
old (it is a stage in the birth of a star).
Thus the Sun was once a T Tauri star but as it is now 4600
million years old, it left that stage of its life a long time
ago.