well velocity is the rate of change of distance with respect to the rate of change of time or ds/dt by definition (more or less)...
so dt can never be zero or we are talking nonsense since we can't divide by zero.
so as far as the validity of your statement
"It doesn't matter how big or small they are, there will always be a time t when v = 0 and a time t' when v > 0." goes the significant measurable thing we are looking at is ds the rate of change of distance. here dt the rate of change of time is just what ever convention we measure time by (hmm relativity excluded i suppose).
but the thing about ds is that if you keep looking at smaller and smaller ds eventually we would get down to the planc scale and uncertainty would kick in. it would be much like the problem Paul Martin discussed with his math professor on the problem of pooring paint into the infinite horn (it's on his website).
well anyway at this point thats my take on it, no charge, no refunds.
regards, tim