Dervan, no. It wasn't explicitly Slovak state, it was common Slovak and Moravian state. Moravia and Slovakia were core of Greater Moravia – Morava is a river in the middle of this empire. Czech territory was conquered later as well as Silesian, Pannonian etc. How could it be Czech state when Czechs fought against it with help of Franks? :)

If Constantin and Methodius were Greeks or Slavs, I don't know. Their origin is not my field actually, I'd rather like to hear some theories from you and other South Slavic members. ;)