What does Bogdan mean?
Bogdan means “given by God” in the Slavic languages.
What does Zmey mean?
Russian/Slavic zmey (feminine zmeya) 'serpent', the inherited Indo-European word for snake that in Slavic folklore denotes a flying serpent, hence 'dragon'. Note: the 'dragon' sense is likely secondary to the original 'snake'; it functions more as a creature-word than a personal name, so standalone given-name use is a modern/stylized choice.