I would like to make sure that I correctly understand how the Sack Objectives in On the Doorstep are supposed to work. I used the search function, but I couldn't find a question that exactly matched what I would like to know.

A sacked character cannot trigger effects. However, the FAQ only mentions triggered abilities. Am I correct in assuming that these terms are synonymous? If that's the case, then only abilities printed on characters that start with a bold trigger word such as "Action:" should be affected.

Given this, can I exhaust a sacked Bilbo to claim the Troll Key Objective? This seems a little unthematic, and I wonder if it was intended this way by the designer.

I would say yes, because you're triggering an effect on the Troll Key, not the Sacked character. Keep in mind that Sacked keeps characters from readying, though.

And for the record, you are correct about what "triggering an ability" included. You can't use Sacked characters' Actions, Responses, etc, but passive abilities still function; they aren't "triggered," after all.