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?

Regarding the first question, you have got some good answers already. (You may want to check out the following threads, too, as I think they align with your thinking quite well: $p$-value: Fisherian vs. contemporary frequentist definitions, Does $p$-value ever depend on the alternative? and
Defining extremeness of test statistic and defining $p$-value for a two-sided test).
Regarding the second question, you start with a Fisherian-esque interpretation by only looking at the null hypothesis and ignoring the alternative. This is pretty intuitive to me, but it does not seem to be fashionable anymore. However, you differ by suggesting to only look at the density at or around the test statistic – not the integral of all densities that are at most as high. The latter would be the Fisherian $p$-value*. It has the advantage of using a unified (and I think reasonably intuitive) scale between 0 and 1 which is based on ranking all possible test statistics from the most extreme (in the Fisherian sense, i.e. having the lowest likelihood*) to the least extreme (having the highest likelihood). Meanwhile, your approach would use a different scale for different tests (Student-$t$, $\chi^2$, $F$, ...), so we would have to develop intuition for each of them. While we can now quite easily judge a value between 0 and 1, there we would be dealing with all kinds of values on quite different scales – not quite as easy.
*Not everyone may agree on that; I had some discussion about it somewhere else on this site. Too bad Fisher is not here with us anymore to elaborate on his position.
Some further references:

*

*Keuzenkamp & Magnus "On Tests and Significance in Econometrics" (1995)


*Lehmann "The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?" (1993)


*Christensen "Testing Fisher, Neyman, Pearson, and Bayes" (2005)


*Spanos "Probability Theory and Statistical Inference: Econometric Modeling with Observational Data" (1999) Section 14.5; there is a newer edition, too.
