Couple questions.

If i put the belt on Alyosha Kraven, does the lion token gain the stat bonuses too? I am guessing not, but thought i'd ask.

Can i assign the belt to the Lion token?

No, the lion doesn't get the benefit of the belt : he's a different character from Alyosha. And no, he cannot be assigned the belt since he only join your force when you place your force on the map : the ressources are assigned during force construction (and the lion does not exist yet at that point)