If I have a Weapon in my hand that gives an attack bonus because of an ability in its text box, will that bonus count for Six-Gun’s ability?
No. You can scrap a Weapon only if it has a + number in the orange stats box.
Interesting Sergeant Six-Gun question. If I have both of his Upgrades on one character, and say an Armor in his Armor slot (so in this case Six-Gun’s Upgrades are in Weapon and Utility, respectively), can I scrap said armor to put on a Utility? One of his guns says you can put in Armor or Utility slots but I'm not sure if you could scrap one to play the other.
No. You choose the slots when you attach the upgrades to the character.