Andor's Trail

Can Gauntlets benefit from Way of The Monk and Unarmed Fighting? It seems reasonable because they feel nothing like weapons but extensions of fists.

No, it's considered a weapon, not a glove and monks can't use weapons.
And let's be honest, you're not really a monk. You're a regular player with a skill with a really fun name.

Then what proficiency do they benefit from?

And can they use Specialisation Two Handed Weapon?

Also, is Kazarite Cloak light or heavy armor?

Light armor