This is similar to a Galton-Watson process (if it is assumed that the production of offsprings are independent).
And assuming that the "parents" die once they give their offspring, we have , where is an iid sequence with mean and variance
And you're looking for and (where - or ? - is 1)
It's easy to compute (using basic properties of conditional expectations), and it gives the result you got by taking the expectation again.
And for the variance, I suggest you use the law of total variance : Law of total variance - Wikipedia, the free encyclopedia
If you have trouble, let us know