The way I understand it, once you attack with your spear (one-handed or two-handed, doesn't matter), you spend your ki point and make 2 unarmed attacks as a bonus action. So yes, to your first question. Flavor-wise, maybe you use your tail and a kick. Heck, head-butt 'im!
Monk martial arts lets you use Dexterity or Strength for your attack and damage rolls, so pick whichever modifier is higher. As for advantage, Kaiakas's vow was said to grant one for 1 minute...
... but I think we're reading the Vow of Enmity wrong. I think only Kaiakas gets advantage.