Re: singular noun after their
"Their" refers to the owner, not the object.
One person can have one thing. The student forgot her book.
One person can have multiple things. The student forgot her books.
Two people can jointly have one thing. Ted and Alice enjoyed thieir vacation.
Two people can jointly have multiple things. Ted and Alice like their neighbors.
There is no need for his/her/their to match the noun that follows.
Also note that "their" is very commonly used for a single person when the gender is not known.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.