Yes, Mitte April and Ende April are both fine.
Ich glaube, das Buch wird Mitte April noch nicht da sein; am besten, Sie kommen Ende April nochmal zurück. "I think the book won't be there yet in the middle of April; it's probably best if you come back at the end of April."
Not really like french (ok, I am going to be annoying, it's very subtle). The correct original french expression is "début de +month". Début d'avril / Fin de mars => beginning of April / End of March. But in French today, "de" can be omitted. Début avril / Fin march. But Début is still associated to "beginning of" and not "early". Litterally : "Early in April" => Tôt en avril.
"In the beginning of April" was not accepted. I think it should. Frage: "Wann ist das Buch fertig? "Question: "when will the book be finished? Antwort: "Anfang April" . And even google says: "Anfang" = "beginning". It does not say "early". "Early" would be "früh", Also accoording to google: https://translate.google.com/#view=home&op=translate&sl=en&tl=de&text=Early . "Early" is just a little bit later than "in the beginning".