Nonzero probability for all time: is it certain it will happen? Is there any possible time varying probability function for an event occurring that is nonzero at all times but the probability of the event it describes occurring over eternity be less than 1? 

Yes. Just like infinite sums of strictly positive quantities can converge to a finite value, so infinite products can converge to finite values. All it requires is that $$\sum_{i=0}^{\infty} \log(1-P_i)$$
converges. (which is the logarithm of the probability that "something" never happens)
