I'd love to see an exposition of your third example, 'the rationals can be listed in full exactly once each'. The standard proof of their countability demonstrates that Z2 (the set of ordered pairs of integers; markup got me here) is countable, and then says the rationals are a subset of Z2 so must be countable, somehow, as well. I've always wondered about a function to enumerate the rationals without duplicates.
Consider the sequence 1,1,2,1,3,2,3,1,4... where the ith element (starting from 0) represents the number of ways to write i as a sum of powers of 2, with each power appearing at most twice. Then you can enumerate the positive rationals without duplicates as follows: Q_n = S_n / S_(n+1). So Q_1 = 1, Q_2 = 1/2, Q_3 = 2/1, Q_4 = 1/3, etc. You can get all the rationals by inserting each one's negation after it. Q_1 = 1, Q_2 = -1, Q_3 = 1/2, Q_4 = -1/2, etc. See http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree
