Yes, it's initial rate, and the assumption, as you correctly suggest, is that (at least over the time before the iodine colour first appears) the rate is constant. In order that the time should be inversely proportional to the rate, the amount (number of moles) of iodine removed by the thiosulfate must be the same in each case, therefore the amount of thiosulfate must be the same. Assuming that you have a stock solution of a given concentration, that means the volume you add must be the same (moles = concentration x volume).
The quantities of reagents are chosen so that the thiosulfate runs out while the extent of the iodine-producing reaction is still relatively small; the concentrations of the reagents in this reaction haven't changed much and the rate is approximately constant over the time t. Theoretically, the inaccuracy ought to be the same for all reactions - except that for very fast reactions, the uncertainty in your measurement of t increases the inaccuracy.