It depends. If the platform or method you use for sending has some sort of notification feature, then they might be able to tell. But in many cases, they won't know unless you tell them.
Generally, it's unlikely. But if you have some kind of special interaction or communication related to the story with the person, they might suspect that you took a screenshot.
It depends on the platform and its privacy settings. Some platforms might not reveal this information to the story owner, while others might have certain features that could potentially allow it.
It depends on the platform. Some apps might have features that allow the creator to see who viewed their story, but not all do. For instance, on Instagram, you can view someone's story anonymously unless they have special settings enabled.
Generally, people won't be aware when you mute their story. Social media platforms are designed to keep such actions private to respect user preferences.