Most sentences are usually put the other way round: the first part (X) is in Nominative, the second one (Y) in Instrumental (because it's Instrumental - Narzędnik, not Miejscownik = Locative). And this one could be written that way as well: Kwiecień jest następnym miesiącem. The question is what kind of 'question' we want to answer: the original sentence talks about 'what is the next month', while my sentence talks about April.
And as usual I'm gonna do some self-advertisement and recommend my guide here, especially part 1,5.
"Następny miesiąc to kwiecień" would actually be probably more natural, but the course creators want to show a variety of options.
No, not really. It's kinda like "every square is a rectangle, but not every rectangle is a square". In this way, "(every) April is a month, but not every month is April". On the one hand, you have "next month" here, but it still would be very weird. When you put something into Instrumental, it kinda 'defines' the other part of the sentence.
"Następny miesiąc to kwiecień" would be perfectly okay.
I wrote something more about it here, part 1,5.
I find it helpful to liken this to the English active and passive. April (subject of the sentence) is next month. (Active because the subjet precedes the verb.) Next month is April (Passive because the subject of the sentence follows the verb) So, put the polish sentence in the same order Noun (subject) verb then the rest. Kwiecień jest następnym miesiącem. Now swap Kwiecień and następnym miesiącem. It doesnt always work that simply with other polich verbs, they get conjugated differently in the passive, but you should get the picture.