Assuming that the velocity of money is constant, does the rate of money growth have to equal zero in order to have zero inflation?
What you are talking about here is the quantity theory of money. One way to state the basic equation of this theory is to say that
Money growth + velocity growth = inflation + Real GDP growth
If we accept this equation, we can see that the rate of money growth does not need to equal zero. If there is growth in the Real GDP, we can have money growth without having inflation. This is because you have said that we will assume no change in velocity. If that is the case, the equation is
Money growth + 0 = inflation + RGDP growth
If we have money growth, inflation can be zero so long as RGDP grows by the same amount as the money supply.
Therefore, it is not necessary that the rate of money growth be zero if we want to have no inflation.