It does seem as though the audio started an idea too late, producing a sharper onset than normal but I don't hear a "k". If anything, it sounds more like a glottal stop to me (which shouldn't be there either but it doesn't change the meaning). Besides, kruf doesn't exist :)
It's not Sie (which would be polite "you") but sie (lowercase), i.e. "them" or "her".
Accusative case, because it's the direct object of rufen -- we don't call "to her", we call "her".
ihr would be dative case, "(to) her".
English doesn't have separate dative and accusative cases; they merged into an objective case, but German keeps them separate. So "her" can be sometimes ihr (Ich gebe ihr ein Buch, I give her a book) and sometimes sie (Ich sehe sie, I see her).
Yes, Duolingo’s correction engine doesn’t care about capitalisation. In reality though, capitalised Sie would be incorrect here because that would be formal you. And that would not only be a mistranslation of the original, it also be grammatically wrong because the imperative uses the informal form. The formal version of “call her” would be: Rufen Sie sie. (The formal version of the imperative requires you to use subject pronoun Sie.)
No, it's not dropped. The confusion comes from the ambiguous meaning of English "to call", which can be either rufen or anrufen in German. Ruf sie means "call her" as in "shout her name". To say "call her" as in "phone her", we say Ruf sie an, with the prefix an- moved to the end.
an is one of the so-called seperable prefixes. Certain prefixes split off of the verb if it's in a conjugated form and then move to the very end of the sentence. Other prefixes stick to the verb no matter what, and still others come in a seperable and a non-separable variant. Usually you can tell from the stress whether the preposition will split off or not. If the emphasis is on the prefix, it will seperate, if the emphasis is on the verb, it won't. Confer this video as well as this one which is the introduction to a whole series going into detail with a lot of different prefixes for more information.
This depends on whether you address a single person (du→ruf) or a group of people (ihr→ruft).
You can check these forms yourself: http://conjugator.reverso.net/conjugation-german-verb-rufen.html, IMPERATIV PRÄSENS is the relevant column.
The reverse, "Ruft sie" and "Ruf den Hund," are also valid sentences; Duo happened to choose the versions you listed in its prompts. The difference is that "ruf" is a command to one person and "ruft" is a command to multiple people, but of course we translate both to "call" in English.
If you were given an English prompt like "Call (somebody)" to translate to German, you could use either "ruf" or "ruft"; Duo will mark both as correct.
If the task was to translate from English to German, then yes, the singular and plural versions should both be accepted. Is it possible that it wasn’t such a task in your case? Could it have been a “write what you hear” type task? In that case you have to write exactly what it says.