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?

Likelihood is a pain to work with.
For starters, likelihood can only go down as you collect more data. Indeed, if your alternate hypothesis is "My coin is biased in favour of tails" and you flip it 0 times, the likelihood of getting zero heads is 100%! Actually doing the experiment can only ever hurt your likelihood value from there.
Second, likelihood is non-exclusive. The likelihood of zero heads from zero throws is also 100% under the null, as well as under the hypothesis "It is a magic coin that always lands how I want." Even if we actually flip our coin and get tails, we're scoring a 75% likelihood. But the null hypothesis still scores a 50% likelihood. It isn't a straightforward battle between null and alternate.
Lastly, likelihood calculations are incredibly sensitive to the formulation of the hypothesis. In practice you often don't know ahead of time (and stating your hypothesis ahead of time is important!) what exact numbers to put in. If you suspect someone is cheating with a coin biased for tails, it's a pain to have to say whether it's a 70% tails or a 75% tails coin. Now suppose you take that coin and flip it 1000 times, getting exactly 748 tails. Under the 75% hypothesis your likelihood of that result is an astounding 2.9%, but under a 70% hypothesis it would be 0.01%. Under the hypothesis we really want, "It's biased to somewhere in the 65-80 zone" it's just messy to define and calculate.
The intuitive measure you're after is "Probability of our hypothesis." Unfortunately, it generally is not a value we can actually calculate from our experiment. It depends too much on things like priors (Fair coins are more common than magic coins, but how much?), and the huge space of possibility of other hypotheses (e.g. what if the coin is fair but the flipper is trained?).
So, we make do with p values as our first filter. To be clear, a tiny p value does not mean that the alternate hypothesis is true. All it means is that the researcher has done enough work to beat the trivial "It's a fluke" standard that they've earned the right to have their work examined.
