Translation:They want to send their children to another school.
This has almost exactly the the same meaning in English, but there is a shade of difference. In Duolingo it is always wise to offer the closest translation. In this case "una scuola diversa" or "una scuola differente" are available to provide the meaning you suggest, so "un'altra scuola" is really best understood as "another school."
Yes, but it is not mandatory. There are some fixed expressions, like "ad esempio", and in many stylistic guides you use it only to avoid to have the same vowel twice in a row, like "ed ecco" or "ad aspettarti".
But, if you remember to use it only before a vowel, if you want to do so, it's always ok.