This is a process called trepanning, and it exists in our world as in Lyra's.
It involves drilling or scraping a small hole in the skull. Evidence of the process has been found from Neolithic times onwards. Cave paintings suggest it was believed that trepanation would cure epileptic fits, migraines and mental illnesses, although Lee tells Lyra that the Tartars believe it will allow them better communication with the gods, and that it is a great privilege to have it performed.
The Yenisei River is the biggest river system flowing into the Arctic Ocean. It flows through Siberia.
The doctor is possibly referring to hypothermia, one of the symptoms of which is mental confusion and hallucinations. The most common symptom is known as "paradoxical undressing", whereby the sufferer believes that their body temperature is uncomfortably high; in an effort to cool down they remove their clothes and freeze.