I cast a Damping Engine and then an Angel of Jubilation. Can my opponent sacrifice a creature to ignore Damping Engine's effect? (Does that count as an activated ability?)
Paying for Damping Engine to ignore its effect is a special action. An activated ability is an ability in the form of [cost]:[effect]. If it does not have that specific template, either on the card or in the rules for the keyword ability, it's not an activatd ability. So your opponent can sacrifice creatures just fine for the Engine.
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)
Activated abilities: rule 602.
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules