Why is the p-value defined the way it is (as opposed to a more intuitive measure)? As I understand it the p-value for some hypothesis H and some test-statistic T is defined as the probability of observing a test-statistic T' which is at least as extreme as T, conditional on H being true.
First question: should "at least as extreme" always be interpreted as "the absolute value of T' is at least as large as T"? If not, when shouldn't it be interpreted that way?
Second question: What is the reasoning behind defining the p-value this way as opposed to just the likelihood of the data -- "probability of observing test-statistic T, or a value in the epsilon region around T, conditional on H being true "-- which seems more clearly related to the strength of evidence that T provides with respect to H than the p-value?

In case you're asking for intuition, rather than for mathematical detail...
For your second question: I find it helpful to interpret the p-value as a percentile, not a probability. A p-value of 0.02 means "If $H_0$ were true, the test statistic value $T$ that I observed would have been among the top 2% possible values of $T$ that are least like $H_0$ and most like $H_A$."
For your first question: It depends on your alternative hypothesis. If you're testing whether a new drug does better than control, then perhaps you've got a one-sided $H_A$, such as "the difference in means between the treatment group vs the control group is a positive difference." If so, then you'd only reject $H_0$ if your test statistic actually points in that direction. You wouldn't want to take the absolute value of $T$ because it would not match your scientific question -- you're not interested in finding drugs that do worse than control. Again, you want to know: "Out of all the possible values of the test statistic when $H_0$ is true, is this among the top X% of values that look most like $H_A$ and least like $H_0$?"
