Israel is part of the middle east, given it's history as being solidly within the region (It's only been a few decades since the area has been majority jewish, it used to be mostly muslim). It's sort of like Turkey in that way. Turkey is considered as part of the middle east since even though there's a pretty solid case for them being considered european, They've just had too much history as a middle eastern nation with the practical annihilation of christian heritage in the country.
Even then, it definitely is. It's Semitic, so closer to Arabic culture than Iran. It's an Abrahamic faith, it's country doesn't unyoke faith and government well at all, and it's also crap to live in (also the living standards change greatly in the Middle East - you have countries like Syria and Yemen, but also Qatar and Bahrain.).
From what I can tell (I've never been to Israel, but I'm a pretty big israel fan) it's sort of like the situation in the US. Yes they are technically secular and have separation of church and state, but it's a little bit of an awkward separation given that basically everybody shares the same faith.
Not everyone will say this, it all hangs on who you ask.
It's pretty much a simple fact, Greeks make up a significant majority in Cyprus (60%), and this isn't even mentioning the fact that Cyprus is actually part of the European Union, a pretty clear argument for it being part of "Europe".
