Can a sword cut a bullet?

Theoretically, yes. Bullets have been fired at swords, and yes, they were cut in half or shattered. Problem is they don't deflect much and now you have multiple bullet fragments hitting you, likely doing more harm.

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What is the sharpest sword in the world?

Damascus swords - sharp enough to slice a falling piece of silk in half, strong enough to split stones without dulling - owe their legendary qualities to carbon nanotubes, says chemist and Nobel laureate Robert Curl. Why do samurai carry 2 swords? All samurai carried two swords as both their badge of office and backup weapons, while ashigaru (peasant soldiers) carried only the shorter of those two swords (wakizashi). In earlier periods the longer samurai sword would be a tachi, later the katana.

What is a Daito sword?

The word daito is a more generic term for "long sword". It is mainly used to refer to the long sword, as counterpart of the short sword called wakizashi or Shoto (?? > Small/short sword). Is Samurai armor bulletproof? Bullet resistant armours were developed called tameshi gusoku ("bullet tested"), allowing samurai to continue wearing their armour despite the use of firearms.

Correspondingly, what do samurai wear under their armor?

Underneath his armour the Samurai would wear a one-piece baggy garment, a kimono on top of that and loose pants.