Translation:Who is Russia's head of state?
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".
Putin is the head of state (President) and the head of government is the Prime Minister, Medvedev.