Could you also theoretically say "государства России" or does it have to be "государства /в/ России"? Is that an idiomatic thing?
Yes, it's acceptable. It has the same semantic difference as "the head of the state in the USA" vs "the head of the USA state".
As a native speaker, I would never, ever say "the head of the USA state." It would be acceptable to say "the US head of state" or "the USA's head of state."
Putin is the head of state (President) and the head of government is the Prime Minister, Medvedev.
Actually VVP is everything. Dmitry is just being used for the rotation of the positions.