Sentence

I'd be tempted to use "they are", or "they're" afraid of you, rather than the s/he's

It's used and may be useful, but it's awkward and does not really sound natural. It's all part of English not having gender-neutral pronouns. You could say "How do you feel when a woman or a child shows fear of you?" or "How do you feel when a woman or a child seems to be afraid of you?"

The cheat code is to use the plural, which is gender-neutral. Either way, you'll be understood, and youre grammar will be demonstrably better than president bush's.

Again we are trapped by the difference between "descriptive" grammar (spoken language) and "prescriptive" grammar (teachers' language). The rule is causing this discussion is one of the legacies we inherited when English grammar rules were forced to conform to Latin. English is a Germanic language, but Latin was considered more "proper".

One is not supposed to end a sentence with propositions, either, which makes no sense in English. Winston Churchill's response to that rule: "That is the sort of nonsense up with which I will not put."