"The cemetery has a church."
Translation:Το νεκροταφείο έχει μία εκκλησία.
Since "a church" is the direct object here and the word begins with a vowel in Greek, should it not be accusative μίαν εκκλησία? (I know this is minor, but just when I thought I was beginning to understand all the case markings, I wrote it like that, and it said I was "Almost correct").
Μία does not become μίαν in the accussative (it does only in archaic sophisticated speech). The -ν in accusative has become obsolete in Modern Standard Greek (that's why we don't say την θάλασσαν anymore, at least not colloquialy, in a more sophisticated-archaic speech you can and it's correct). Μια (without accent) can become μιαν (without accent) in everyday speech when followed by vowel. Μιαν is an accepted answer.
You may have got the gender wrong -- if you type έναν εκκλησία, that's wrong, but rather than correcting the article, Duo assumes you got the article right and the noun wrong.
As for the other question, ναός seems to cover both what we'd call "temple" in English and many cases of what we'd call "church", at least Greek Orthodox churches.
I'd say that's a fine formulation. There's an absolutely massive cemetery near us that has a church in the middle, so I'd say that cemetery has a church. If we were talking about a smaller church that had a small graveyard attached, then yes, I'd say "the church has a cemetery".
I was picturing the situation as you describe it here (or as it crops up in any number of English murder mysteries). And, proud of myself for remembering the word for "chapel or church" I learned in Memrise, I wrote, "το νεκροταφείο έχει καθολικό." Does this stray too far or could it be added, do you think?
Just μία; in a sentence like this, that's more like saying that the cemetery has "one" church. If you simply wanted to denote that the cemetery has "a" church, you'd omit the article completely. If you wanted to really emphasise that this cemetery has only one church - as opposed to some other cemetery that has two churches - you could say that it has "μόνο μία εκκλησία"