Do the apperance in the atomic nucleus of virtual matter-antimatter particle pairs play a role in the random nature of radioactive decay?
The random character of the process itself doesn't depend on any specific virtual particles - instead, the random character of all microscopic processes in the world is a basic consequence of quantum mechanics. The existence of virtual particles is a consequence of quantum mechanics, too. Obviously, virtual particle pairs are important - in a particular computational scheme - to calculate the actual decay rate of a particular unstable object.
So the virtual pairs are not the cause of the randomness itself - they're a consequence of the randomness - but in any calculation, the virtual pairs are a part of the cause of the actual numerical value of the decay rate.