It is not possible to launch nuclear missiles at 700 meters as the water pressure would crush them. Normal launch occurs at keel depths of less than 50 meters.
Textual labels throughout the sub are either meaningless combinations of Cyrillic letters or transliterated English words. This is most evident on the depth indicator, where the Cyrillic letters spell the English word "meters" instead of the Russian "metry".
When the submarine surfaces and crashes through the ice, two of the missile launch doors are open, and later are closed. The impact of the ice would have damaged the doors and they could not be closed.
The submarine bridge is equipped with LCD screens (although apparently imitating monochrome CRT monitors), these weren't available in 1984, and certainly not on Russian submarines.
It's been established that the TARDIS translates for the Doctor and his companions - however, there's no explanation given for why the Ice Warrior and the Russian submarine crew can understand each other.
Captain Zhukov's sidearm is a Browning High-Power. It was never used by the Soviet military.
Much like on an airplane - a weapon should never be fired on a submarine.