I don't think the issue here is with the object - in both cases, the sentences have an object. What they do not have is the actual act of physically picking something up (taking it) literally and doing something with it. 

In both of your example sentences, what is being taken is figurative. 

可把我吓坏啦 - This is a very similar type of usage as the English phrase "The sight of that stunning scenery really grabbed me." In this case, no one was actually grabbed. Same deal here. I'm not a grammarian, so I can't explain it to you in those terms, but if you study the actual make-up of the word 吓坏, you'll see why "把" makes sense. These two characters together - 吓坏 - mean "to really frighten". But separately, they mean 吓 - "to startle" and 坏 - "to break". You have to "take" something (把) before you can startle it so bad you break it. 吓坏 is actually an act-result type of word. Because 吓 happened (I was started), then 坏 happened. 

Same thing here - 猪小姐把那个丑八怪的故事仔细地讲给了猪先生 - What is being "taken" is the story (故事), so 故事 is the object. But then what is being done with the story? It's being GIVEN (讲给) to Mr. Pig. You have to TAKE something (pick it up, figuratively have it) before you can GIVE it.