Concerning The Number of Ways of Drawing a Full House vs. Two Pair The Wikipedia entry for "Poker probability" gives the following result for the number of ways of drawing a full house:
$$ \binom{13}{1} \binom{4}{3} \binom{12}{1} \binom{4}{2} = 3, 744. $$
The logic here is conventional: From the 13 kinds you choose 1, then choose 3 of the 4 available cards of that kind. Then, from the remaining 12 kinds, you choose 1, then choose 2 of the available cards of that kind. No problem.
However, for the number of ways of drawing a two pair it gives the following:
$$ \binom{13}{2} \binom{4}{2} \binom{4}{2} \binom{11}{1} \binom{4}{1} = 123, 552. $$
But, by the logic used for drawing a full house, shouldn't it be the following:
$$ \binom{13}{1} \binom{4}{2} \binom{12}{1} \binom{4}{2} \binom{11}{1} \binom{4}{1} = 247, 104? $$
OR, conversely, by using the logic used to get a two pair, shouldn't the number of ways of drawing a full house be the following:
$$ \binom{13}{2} \binom{4}{3} \binom{4}{2} = 2,808? $$
As you can see, this isn't just a matter of inconsistent approaches that give the same result; these give different values. I just don't know how I'm supposed to be able to intuit these sort of things, seeing as they seem like the same thing to me.

The problem with your $247,104$ is that it counts each two-pair hand two times, according to which of the pairs you mention first. But 5H-5D-7S-7H-9D is the same hand as 7S-7H-5H-5D-9D, so it gets counted both with fives first and with sevens first.
In contrast, for a full house it is unambiguous which value is the one with three cards and which is the one with two.
