All current B737NG aircraft are certified for Polar Ops with certain limitations. You did not specify what model of 737 you were referencing?
As for initialization of the FMS it is simply a matter of where and how the certification took place. I do not have the 737 data at hand but in the case of the 777, the highest latitude that the airplane ADIRU can be initialized corressponds to the airport Lat. at Svalbard (Longyearbyen-EBSB), which just happens to N78.14.75 minutes. I would bet that the 737NG is the same as the BBJ's are capable of some Polar flight plans
The FAA has defined Polar Operation to begin above N78. Generally speaking there is no "violent weather patterns in the Polar regions. Snow and whiteouts, yes but the area is some devoid of the weather systems that can be found at lower latitudes. Operators need special authorization to operated in the Polar regions and that would include familiarity with communication issues, passenger recovery, alternates, QFE/Metric altimeter procedures, MNPS, solar storms, HF blackouts, ETOPS, etc. The list goes on and on.