Learn English – First and third person in second conditional exception

The Three Conditionals (or sometimes Four Conditionals) provide perfectly adequate explanations of the sentences you cite.

So did what was taught in US high schools in my youth under the rubric Sequence of Tenses. (At least I think it did; I wasn't paying that much attention at the time.)

What needs to be kept in mind is that these are what I have elsewhere called ‘baby rules’. Students must master crawling before they walk and walking before they run, and rules of this sort are directed to crawling and walking. Your ordinary English class or English grammar textbook is a linguistic hothouse, where students are carefully isolated from uses which lie outside, and appear to violate, the rules they are learning. This provides a foothold on the language which permits the student to grow in mastery and confidence (and of course ‘self-esteem’, which is of great concern to modern educators) before they are released into the wild and must confront the horrors of real life: colloquial, business and literary English.

What tchrist (indignantly) and Barrie England (charitably) are concerned to point out (on ELU, where the audience is presumed to be sophisticated users) is that these rules are inadequate maps to large tracts of that wilderness; such sentences, for instance, as these:

If you’ll pick up the beer I’ll get the brats.
If you’d ever actually read Lévi-Strauss you wouldn’t say stupid things like that.
If it was me I’d give him what for.
If you actually look it up what he said was completely different.

Rules are for learning with; but once you’ve learned enough to walk on your own you can discard these crutches.

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.