I refer you to the "Potentially historically correct scenarios" thread for more info, but in short:

What we assume about the Mongol Invasion campaign is that the Mongols will be able to establish a foothold in Japan and rampage across the isles. This is not historically correct in the sense that they were actually defeated earlier, in the invasion phase à la Dieppe, but that doesn't mean the MI scenario wouldn't be historically possible. Kind of like the Mongol Invasion *without* the kamikaze.

About units: No-Dachi Samurai actually come into their own here, since they were really used extensively in the late Kamakura period (well, admittedly only after the Invasion), unlike in the Muromachi period where the original STW is set. Warrior monks were also more plentiful during Kamakura, and it's conceivable they would have fought with the Hôjô Bakufu to save the nation.

But I'll let someone who knows more than I to explain the Kensai.

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"The right use of the sword is that it should subdue the barbarians while lying gleaming in its scabbard. If it leaves its sheath it cannot be said to be used rightly."
- Tokugawa Ieyasu: Legacy