Learn English – Where were “should”, “shall”, and “must” in the 18th Century

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.

It's the phrase 'cheek by jowl', which the OED lists under the entry for 'cheek':

5.a. cheek by jowl; †cheek by cheek (In 6–7 cheek(e to jowl, by chole, jole, joll, gig(g by geoul, jowl, 7–8 jig(g by jowl, 9 cheek by chowl, for chowl, and jowl, Sc. cheek-for-chow, dial. jig-by-jow.) Side by side; in the closest intimacy.

It's also listed under 'jowl | jole' (n1):

1.a. A jawbone, a ‘chaft’; a jaw; esp. the under jaw; pl. Jaws.

4. Here perhaps belongs the phrase cheek by jowl, in earlier usage cheek by cheek: see cheek n. 5a. In this the j form is known from 1577, which is somewhat earlier than it is known in sense 1 above. The 17th c. variants cheek by chole, chowl, agree in form better with jowl n.2 or . But it is probable that, by the time the phrase came into use, all three ns. were already felt as one. The following examples supplement those under cheek n.

and that refers to 'jowl | jole' (n2):

The external throat or neck when fat or prominent; the pendulous flesh extending from the chops to the throat of a fat person, forming a ‘double’ chin; the dewlap of cattle; the crop or the wattle of a bird, etc.;

So the OED seems to be uncertain as to whether the 'jowl' in 'cheek by jowl' refers to the jaw, the neck, or the roll of fat which makes a 'double chin' - or even perhaps 'head'.

But 'walking cheek by jowl' definitely means walking very close together.

This is from The School of Manners, I'm guessing?

Walking with thy Superior in the house or Garden, give him the upper or righthand, and walk not just even with him cheek be joll, but a little behind him, yet not so distant as that it shall be troublesome to him to speak to thee, or hard for thee to hear.

So Garretson is saying, when you're walking with someone more important than you, don't walk side-by-side with them as if you were their best buddy, instead walk a little bit behind them (but not so far behind that it's a pain in the arse for them to talk to you).