Suppose avg(A) and avg(B) are normally distributed. This will be exactly true if A and B are normally distributed, because a sum of normally distributed deviates is itself normally distributed. It will be approximately true for large samples however A and B are distributed, by the central limit theorem.
Yes if you accept that the independent assumption.
Reminder: As what Ichbin mentioned above, the variance of the ratio is usually
large if the random variable in the denominator have a high probability to
fall in the neighborhood of 0. So make sure, if you have make any distributional
assumptions, check whether the moment exist or not.
that ratio should follow the ratio normal dist. See this paper: D. V. Hinkley (December 1969). "On the Ratio of Two Correlated Normal Random Variables". Biometrika 56 (3): 635–639 and substitute the corresponding parameters. If the means are zero you should Cauchy dist even the variances of the A and B are not one. Also Cauchy dist is a special case of the ratio normal dist.