Is Jenny (Дженни) a name used regularly in Russia, or is it used here for the benefit of English speakers? I'm curious, since my name is Jennifer.
I don't think so. From experience, I've found most female Russian names end in а or я.
Does Russian distinguish between "yet" and "still"? Could уже be translated as "still" in this example?
Дома is already a set phrase that means "at home". I'm pretty sure that adding в would mean "in at home".
Edit: According to this дома is now considered an adverb rather than a noun, therefore it's no longer part of the normal declension system. Interesting side note: Latin domo underwent the same change from an archaic locative case to an adverb, and in German, zu Hause saw a similar process.
Можно сказать "в доме", но это будет означать в здание. "В дома" - это ошибка в любом случае.
English also sometimes uses "home" as an adverb, so the best translation for this sentence would be, "Jenny is already home."