To the above list I add whose vs who's
"Whose shoes did I trip over in the hallway and who's going to pick them up?"
The first is the possessive, the second the contraction of "who is".
as in: "He literally laughed his head off."
No, he bloody didn't! Or if he did, he must be the first person in history whose head was rolling on the floor detached from his body after a joke -- without some outside intervention.
With regard to they, them and their used as a gender-neutral singular pronoun, the matter does have historical literary precedent -- Jane Austen certainly used the convention. It seems acceptable to me, especially now, when, with the increasing use of gender-neutral nicknames, to presuppose a gender could be misleading, but to use 'it' is blatantly incorrect. S/he, him/her, his/hers is cumbersome, and sie, hir, and kir contrived and artificial. "They" is, if not perfect, certainly the least troublesome way to indicate that the person being referred to is of an unknown gender.