AnonymousI think the modal 'would' can be used as a past form of the modal 'will'. Can we use the modal 'should' as a past form of the modal 'shall'?
Yes, but probably not the way you're thinking of it, to judge by your example.
would is the 'backshift' (past) of will. Note the following examples of reported speech.
Georgina says that she will attend the meeting.
Georgina said that she would attend the meeting.
The relation between shall and should is like the relation between will and would in conditional patterns:
If Bernie passes the exam, I shall be very surprised. [very formal substitute for will in AmE]
If Bernie passed the exam, I should be very surprised. [very formal substitute for would in AmE]
________
In the case of paying for the pizza, He should pay for the pizza is present, not past: It is expected of him to pay now or soon. I believe it is his duty to pay now or soon.
The past is should have. He didn't pay for the pizza, but he should have paid for it. It was expected of him to pay then, at that time. I believed it was his duty to pay then, on that occasion.
________
Anonymous It wasn't fair for him to not pay for pizza (not to pay for pizza??).
It wasn't fair for him not to pay for the pizza. He ate the pizza, so he should have paid for it too.____
In short, should is only rarely used as a past form -- in very formal writing and typically in conditional sentences. In 99.9% of the cases you encounter where you want the past of should, should have is the more likely choice. And should have always means didn't -- was expected to (do something), but didn't (do it).
CJ