I find this translation misleading, because "May he use her restroom", as in "Is he allowed to use her restroom?" would be in German: "Darf er ihre Toilette benutzen?"
However, "Dürfte er ihre Toilette benutzen" would be equivalent in English to: "Would he be allowed to use her restroom?"
Dürfte is Konjunktiv 2, which means that it expresses an hypothetical situation, much like in English with the words "would" and "were", the only difference is that in German it is possible to use Konjunktiv 2 using only one word that has been "tempered" with, usually, but not exclusively, in the Präteritum form.
Er hat = he has. Er hatte = he had (past - Präteritum). Er hätte = he would have.
From this we can surmise that:
Er darf = he is allowed/he may. Er durfte= he was allowed (past - Präteritum). Er dürfte = he would be allowed.
Not a native speaker, but my understanding is dürfte is a conditional version of darf. It is tricky to translate them differently into English, but its like you're not asking if you're allowed to do something in the sense of it being against the rules, but rather just politely asking permission.
Surely for a simple request this ought to be 'darf er ihre Toilette benutzen?' 'Dürfte er ihre Toilette benutzen?' is Konjunktiv 2, i.e. conditional, and would mean 'Should he use her Toilet'? There must be a distinction in sense between a present indicative and a conditional, unless, despite the tense, this is just the way a German prefers to say it. Any native German speaker out there?!
as a child, i am expected to use 'may' because it is polite, therefore, it sounds completely normal to me. i assume that as you get older, you will use it less because one on the main places it is used is in school when talking to teachers, and at home when talking to your parents or guests.