Are "might" and "should" past tenses of "may" and "shall", respectively?

Both of the past-tense examples sound somewhat archaic, but that is to some extent because the use of shall in the present-tense sentences does also, imo. (The use of present-tense may in the second sentence sounds formal, but not archaic.)

Probably the last person I heard use might and should with these specific meanings was my grandmother, who learned English as a foreign language about 100 years ago (literally). If your questions is whether you should :-) use the words with these meanings, I would say no, as it will simply be confusing. To convey the past-tense sense of these, you might have to come up with workarounds:

When he was at school, he was not allowed to go to the bathroom ...

He thought that they would go to the restaurant.

Hmm, that second one is tricky, but it's the result of substituting the non-archaic will for shall in the present tense.

Contrary to many of the other answers, many of the modal verbs that are listed in the dictionary as the past tense of another modal are in fact still used commonly used in that way in modern English - but sometimes only in certain senses. But the other answers are spot on that correct use of modals in English is a minefield for the learner :-)

• May/might: The two most common uses of may are to indicate permission ("I'm about to tell the students they may ask questions after the class") and possibility ("I think it may rain later"). While it's technically correct to use might for the past tense of both, only the possibility sense works well: "I thought it might rain later" is identical to its counterpart apart from the tense, whereas "I was about to tell the students they might ask questions after the class" sounds slightly odd because the "unlikely possibility" sense of might comes to mind first - and, while the context here makes it clear that's unlikely, the reader will likely have to pause to figure that out. This difference between the meanings is why example 1 in the question sounds a little strange.

• Can/could: This works in all senses - for example: "I think I can do it" -> "I thought I could do it"; "I'm told I can have one" -> "I was told I could have one"; "He can't have eaten it all by now" -> "He couldn't have eaten it all by then".

• Will/would: Again, this works in all senses: "I know someone will take the job" -> "I knew someone would take the job"; "These days he will lose his temper at the drop of a hat" -> "In those days he would lose his temper at the drop of a hat"; "I think that will work" -> "I thought that would work" etc.

• Shall/should: This is largely the exception to the rule - use of should as a past of shall is somewhat archaic, largely because shall is, itself, archaic in most uses. In practice the "obligation" sense of should dominates. For instance, "You shall go to the ball" tells Cinderella emphatically what is about to happen, whereas "The fairy godmother told Cinderella she should go to the ball" implies that Cinderella was told of an obligation upon her rather than a prediction of her future. Of course, this is why there seems to be a meaning shift in example 2 from the question.

  The one sense in which should does almost work as the past tense of shall is the use of shall for commands ("You shall leave!" -> "I told them they should leave"), though they are not equivalent in tone (the former is a very forceful command given with confidence that it will be obeyed, whereas the latter gives a sense that it is better to leave than not to, but that there is some choice involved in the matter).

(In producing the examples I've used the definitions from Merriam Webster - I've tried to cover the common definitions, but I've ignored some of the rarer ones, and some examples span more than one definition.)

Update: Note that I've used a "reporting" or "reported speech" structure in the examples, because using the past forms of the modals there in the way I've described is idiomatic in both formal and informal contexts, in all the dialects I'm familiar with. Some might argue that this is not a "true" past tense, but an artefact of the way we construct "reporting" clauses (because, for example, in the sentence "I promised that I would come and I will come" the action of "coming" is entirely in the future). However, exactly the same usages are possible with the modal verb in a main clause (though this is more common in written than spoken English). For instance: "Later today I will find out my result" => "Later that day I would find out my result"; "Now I can jump three yards" => "Then I could jump three yards [but I can't any more]".

The dictionary is technically correct (as might be expected) but you should consider that informal english does not necessarily conform to the proper and exact denotations.

In the example above, "might" is used for the past-tense construction of "may" in a manner which is perhaps more familiar to persons who are not in the habit of reading Dickens. Mike Pope is correct in asserting that few people currently are in the habit of using the proper past-tense constructions of those words.

Off the top of my head, the only use of the word "might" in reference to the past-tense of "may" rather than as an indicator of probability or the completely unrelated denotation of strength is in the (archaic but still current) poem "Star Light, Star Bright" where it is used to rhyme with "night."

"Should" is a similar case; it is somewhat conflated with "shall" but with the same probablistic connotations that "might" is burdened with; most uses that I see in current language are (unhappily) people applying wishful thinking to functioning of processes, e.g., "Rebooting the computer should fix the problem."

It is still proper to use these words in their appropriate functions, I think, provided that you make clear from the context your intended meaning.

It depends. Technically yes, but "should" and "might" are also the subjunctive form and imperfect forms of "shall" and "may". They can be used as conditionals, too, which is basically the subjunctive form in essence. Here are some examples that one could argue their being the past tense forms of "shall" and "may":

"I should have gone to the movies." "I shall go to the movies."

"I might have known the answer." "I may know the answer."

There are other ways to show this. Technically, the past of "will" is "would" and the past of "can" is, get this, "could". No way lol! The only other very common modal is "must" and its past tense is a pathetic "must" so no change. There are other modals besides these, but they are usually considered "semi-modals" because they can be used as normal verbs, too.