>>the possessives / pronouns referring back to everybody MAY also be plural,
This part is true,
>>in which case the verb would be used in the plural, too.
but this part isn't. Notice that there aren't any plural verbs in the examples you quote. 'Had' can be singular or plural, but see what happens when you change the tense:
>>When everybody had finished eating, the waiters took away their plates.
When everybody has finished eating, the waiters will take away their plates.
Think about it: Everybody *are? Everybody *have? Everybody *do? Surely Swan didn't say that.
>>Is it correct to say everybody knows + Plural (their ...)
In conversation or casual writing, yes. In formal writing, it's better not to.
Either way, it's often better simply to rewrite the sentence:
All employees know their responsibilities.
We all know our responsibilities.
Okay, everybody, let's get our backpacks ready!
Hey everybody, time to get your backpack ready!