The pgf of (or , equivalently) is easily found to be .

Then you probably know that the pgf of is , etc., in general . The formulas from the text can then be verified by induction; it's just tedious computation.

As for (2), your computation of the extinction probability is correct. When , the probability is 1, hence conditioning to extinction is like doing nothing: ( being the pgf of conditioned to extinction). When , this probability is positive ( ). In this case, you can deduce the pgf of the offspring distribution of the process conditioned to extinction: from your lecture (very likely), it is . As you notice, this is the same as except that and are swapped. The pgf is therefore obtained by swapping and in . Notice that this swapping resumes to the situation when , where the extinction is almost sure: that's consistent with the fact that the process conditioned to extinction reaches extinction eventually.