Learn English – tenses agreement, and the usage of the second conditional: “Would it be okay if I didn’t (don’t) come to your show next week?”

Start by dismissing the ‘first / second / third conditional’ notion from consideration—that is a ‘baby rule’ for introducing beginners to conditional constructions.

The sentence given in your quote may mean two different things: it may be (1) a tentative (hypothetical) conditional in the present tense with future reference, or (2) an open (possible) conditional in the past tense with future-in-past reference. This ambiguity is resolved in the discourse context.

• (1) If you asked me nicely, I might get you a drink; but right now you are behaving very rudely and you can get your own. —Here the past-form asked and might do not imply past reference but ‘modal remoteness’—that is, you are doubtful that the condition and its consequence will be realized.

• (2) I told you then that if you asked me nicely, I might get you a drink. —Here the past-form verbs do imply past reference; the sentence represents backshifted If you ask me nicely I may get you a drink, which implies that at the time of utterance you regarded the condition and consequence as future possibilities.

• Note, by the way, that the pre-backshift present-tense version may equally well be expressed as If you ask me nicely I might get you a drink if the MAY in the consequence clause represents possibility rather than permission. In that case, it will retain the same form, might, when backshifted. The use of modals is very complicated: it has been constantly changing for hundreds of years, and we are in the middle of what appears to be an acceleration of the change.

Your rewrite, however, will bear only one interpretation, a past counterfactual:

• If you had asked me nicely, I would have got^† you a drink. —Here the ‘double past’ in both verb constructions (had asked, would have got) marks the sentence as both past-tense and modally remote. This is understood to mean a ‘condition contrary to fact’: on the past occasion of which you are speaking both the condition (your asking nicely) and its consequence (my getting you a drink) failed to occur.

^†As user3169 and Lucien Sava point out, this is the correct past participle of get in BrE but must be gotten in AmE.

A conditional sentence has two parts: the "if-part", or protasis, and "then-part", or apodosis.

In the apodosis, we use would to create that "conditional" feeling. So the sentence

If we had had lots of money, we should have travelled round the world.

would be illogical: the protasis clearly says that we did not have money during some period in the past, but the second half of the sentence is not an apodosis, because it uses the wrong modal verb. The second half looks like a normal sentence on its own:

We should have traveled around the world! (instead of doing some other things in the past, we should have traveled around the world)

This clause implies that we had the ability to travel around the world. It contradicts the protasis.

The same applies to your conditional 2 example:

If I worked harder, I should pass the exam. [improbable situation in the present or future],[moral obligation]

The first part invites some apodosis with would. The second part just plainly states your obligation to pass the exam. But according to Wikipedia,

Occasionally, with a first person subject, the auxiliary would is replaced by should (similarly to the way will is replaced by shall). (Wikipedia says this about the apodosis of both the second and the third conditional)

So maybe we can use should there after all, since I is a first-person subject. Let a native speaker decide.

The use of should is usually "deontic" (what should be: used to express norms, expectations, speaker's desire) while the use of would is "epistemic" (what may be).

The modal verb should could be sometimes used in the epistemic mood too, so we can come up with sentences such as

If I worked harder, that should be surprising!

But the effect would be comical and ironical, not the usual effect with the conditional sentences. The two halves of the sentence would still look somewhat disjointed.