Three important issues I don't understand three important issues :
(1) Why do we calculate the test statistic under the null hypothesis ?
(2) Why do we reject  the null hypothesis when :
p-value < significance level ?
(3) Why is type-I error more serious than type-II error ?
 A: 1) 
I found this pre-print paper by @Michael Lew to clarify many things for me.
In terms of calculating a p-value under the null hypothesis, it can be seen as more of a matter of convenience than anything else:

the null hypothesis serves as little more than an anchor for the
  calculation|a landmark in parameter space

To P or not to P: on the evidential nature of P-values
and their place in scientific inference
2-3) 
The original use of "significance" by Fisher grouped results into interesting/not interesting. If the p-value was not less than some threshold (unique to the given experiment determined subjectively) he would attribute it to low signal to noise ratio and ignore the result. There was no concept of type II error as one would never accept a hypothesis. 
The Neyman-Pearson hypothesis testing framework instead determines the significance level via cost benefit analysis (what are the relative costs of Type I and Type II errors) using two (or more) plausible alternative hypotheses. The two approaches were combined into one by textbook writers starting in the 1940s and many claim the result makes no sense. 
The wikipedia page on NHST needs some cleaning up to incorporate the contributions of different editors but it has many good references. Because the method that is taught is a result of historical accident (not reasoning or logic) it will be difficult for you to understand without knowledge of the history.
EDIT:
If we look at Fisher's early (first?) description of the significance test we see that in the case of the lady tasting tea it was plausible that she could not distinguish the order in which tea/milk was added at all. Thus this was a logical null hypothesis. He also states the significance level is "open to the experimenter". 
However later he states that a null hypothesis that the death rates of two groups of animals are equal is also valid. In this case I think he simply missed the logical consequences of choosing an (often implausible; it is highly unlikely two death rates will be exactly the same) null hypothesis that is the opposite of the research hypothesis (why do the study unless you suspect the death rates will be different). These issues are described by Paul Meehl in Theory-Testing in Psychology and Physics: A Methodological Paradox and
Appraising and Amending Theories: The Strategy of Lakatosian Defense and Two Principles That Warrant It
Quoting Fisher:

A LADY declares that by tasting a cup of tea made with milk she can
  discriminate whether the milk or the tea infusion was first added to
  the cup. We will consider the problem of designing an experiment by
  means of which this assertion can be tested.
...
It is open to the experimenter to be more or less exacting in respect
  to the smallness of the probability he would require before he would
  be willing to admit that his observations have demonstrated a positive
  result. It is obvious that an experiment would be useless of which no
  possible result would satisfy him. Thus, if he wishes to ignore
  results having probabilities as high as 1 in 20 -the probabilities
  being of course reckoned from the hypothesis that the phenomenon to be
  demonstrated is in fact absent- then it would be useless for him to
  experiment with only 3 cups of tea of each kind. For 3 objects can be
  chosen out of 6 in only 20 ways, and therefore complete success in the
  test would be achieved without sensory discrimination, i.e., by “pure
  chance,” in an average of 5 trials out of 100. It is usual and
  convenient for experimenters to take 5 per cent. as a standard level
  of significance, in the sense that they are prepared to ignore all
  results which fail to reach this standard, and, by this means, to
  eliminate from further discussion the greater part of the fluctuations
  which chance causes have introduced into their experimental results.
  No such selection can eliminate the whole of the possible effects of
  chance coincidence, and if we accept this convenient convention, and
  agree that an event which would occur by chance only once in 70 trials
  is decidedly “significant” in the statistical sense, we thereby admit
  that no isolated experiment, however significant in itself, can
  suffice for the experimental demonstration of any natural phenomenon;
  for the “one chance in a million” will undoubtedly occur, with no less
  and no more than its appropriate frequency, however surprised we may
  be that it should occur to us. In order to assert that a natural
  phenomenon is experimentally demonstrable we need, not an isolated
  record, but a reliable method of procedure. In relation to the test of
  significance, we may say that a phenomenon is experimentally
  demonstrable when we know how to conduct an experiment which will
  rarely fail to give us a statistically significant result.
...
It is evident that the null hypothesis must be exact, that is free
  from vagueness and ambiguity, because it must supply the basis of the
  “problem of distribution,” of which the test of significance is the
  solution. A null hypothesis may, indeed, contain arbitrary elements,
  and in more complicated cases often does so: as, for example, if it
  should assert that the death-rates of two groups of animals are equal,
  without specifying what these death-rates usually are. In such cases
  it is evidently the equality rather than any particular values of the
  death-rates that the experiment is designed to test, and possibly to
  disprove.

I think a crucial point that others have missed in reading Fisher is that he never specifies a need to specify the significance level beforehand. A reasonable level could be chosen given the experimental conditions that unfolded. Also no strong conclusion about a natural phenomenon should be derived from a single study anyway, a single study could only provide evidence pointing towards the phenomenon's existence, thus suggesting the usefulness of further study. Thus the significance level could be arbitrary and based on convenience (since it could change) but it was not "binding". 
While for Neyman-Pearson a single study could not provide any evidence for the existence of a phenomenon. The "significance" level was definitely non-arbitrary but it was binding if the pre-specified error rate of the decision making process were to be valid in the long run.  Citing Neyman and Pearson the same as @MichaelLew in the above arxiv paper:

We are inclined to think that as far as a particular hypothesis is
  concerned, no test based upon the theory of probability can by itself
  provide any valuable evidence of the truth or falsehood of that
  hypothesis. 
But we may look at the purpose of tests from another
  view-point. Without hoping to know whether each separate hypothesis is
  true or false, we may search for rules to govern our behaviour with
  regard to them, in following which we insure that, in the long run of
  experience, we shall not be too often wrong.

The hybrid approach (which is being taught to researchers) uses a pre-specified arbitrary and binding significance level which is different from both. It neither adjusts the level due to experimental conditions nor involves cost-benefit analysis. 
A: *

*The test statistic is chosen to be a measure of the discordance of the data with the null hypothesis, in some direction of interest (e.g. the difference of sample means between two groups, the correlation between successive observations in time, &c.). The bigger it gets, the more evidence against the null hypothesis.

*Well, we don't always. Sometimes the p-value is used just to calibrate the test statistic: a p-value of 0.03 means that you'd see a test statistic as big or bigger than the one you saw only 3 times out of a hundred if you were to keep repeating the measurement and if the null hypothesis were true. But if you want to say "I reject it" or "I don't reject it" (& to follow some different course of action in each case) then, when the null hypothesis is true, the p-value will be lower than a significance level 0.025 with probability 0.025 (it's uniformly distributed†); thus you control Type I error, the probability of rejecting the null hypothesis when it is in fact true, to 0.025.

*More serious in what way? If you mean that making Type I errors has worse consequences, that's clearly not true in general.
† It's a bit different when you're sampling discrete variables—you'd have to say it'd be lower than the significance level with probability 0.025 or less.
