Does Burrowing's extra damage affect non-attack damage and where inside/outside Tokyo?
According to iello, who wrote the rules for the French and English versions of the game states that you deal extra damage when you are in Tokyo:


  Cost: 5 Energy.
  
  Type: Keep.
  
  Text: "Deal 1 extra damage on Tokyo. Deal 1 damage when yielding Tokyo to the monster taking it."
  
  
  You deal 1 extra damage when you are in Tokyo.
  You deal 1 damage when you yield Tokyo to the monster attacking you.
  


As to the second part, what about damage cards, iello says this about Firebreathing:


  Firebreathing: "Your neighbors take 1 extra damage when you deal damage"
  
  Q: Does fire breating activate when you deal any damage?
  
  E.G. Burrowing "Deal 1 damage when yielding Tokyo to the monster taking it."
  or only when attacking?
  
  A: Anytime you do damage wether by attacking or using an ability.


Under this reasoning, I would assume that Burrowing would deal extra damage from any card with an effect that deals damage.