Normally adverbs used with "listening" would be things like "closely", "intently", "carefully", etc. to show that you are interested. I think something like "I'm listening with amazement" would work, but only in telling a story or writing an essay, as it sounds quite literary. You wouldn't say something like that to someone as they are telling a story.

None of the above. There are many alternatives though e.g. "no way!" "that's crazy!" "oh my gosh" etc etc.

Sorry but I don't consider any of those "natural", unless you were saying them sarcastically. You could say, "I don't believe it" or "Seriously?" if the story is unbelievable.

It's all about context. I would personally say "I'm listening in amazement" but it depends on the context of my sentence. For example: Sara was telling us about her experience at the Boston marathon and I was listening to her in amazement with all of her words forming images in my head. I could not help but listen in wonder to the new professor this morning. Are they usual sentences? Probably not, but they are not unusual. And no one would look twice at you for using them except to think that you're probably pretty smart.