Does 'many a' take a singular pronoun or a plural one?
Many a is followed by a singular noun. So also does it follow a singular pronoun?
There is no one-size-fits-all rule. You simply use whichever pronoun makes sense given the context. Notice how all five of these are different:
Notice that the last one takes a plural verb.
And regarding pronouns, you shouldn’t be thinking of singular or plural. That’s the wrong axis.
In a comment, Chaim insightfully notes that:
On that view, your only plural is #5, so a one-size-fits-all rule consistent with your examples is to use the same number that you would use if the word "many" were elided: an old tree was damaged, a new student comes thinking that they know stuff, a young mother needs, a young father is happy, and a young couple are pleased.