The second part of the word necromancy (-mancy) also derives from the Greek μαντεία, meaning oracle, divination and prophecy. ^.^
(The derived suffixes -mancer (sne who employs a specific form of divination) , -mantic (of a specific form of divination) and -mance (carry out a specified form of divination) originally derive from this word as well.)
Kind of. I think 'τάφ-' (and θάφ) are roots which signify some sort of meaning to with burials. So you've got words like τάφος, for grave, and νεκροθάφτης for gravedigger, and ταφόπλακα for headstone. But I don't think there's an actual 'ταφείο' word, although I could be wrong.
I'll have to say that ταφείο is used rarely with the same meaning as νεκροταφείο when it is a specific graveyard. Ταφείο πολιορκημένων = the embattled's graveyard. I would say that it is more like "burial ground" and can be used for other burial sides than deads' burial sides, but it is a bit rare.
Yes, pretty much, except it's κοιμητήριο, with the omicron. It's not 100% wrong to drop the o, but it's a bit informal. It's derived from the verb κοιμάμαι and is fairly obviously where we get cemetery from in English.
In my experience, νεκροταφείο is way more common though. I don't think I've ever heard a Cretan refer to a κοιμητήριο in speech, and all the signs for cemeteries say "νεκροταφείο".