PC and Others

Context:

Man held over PC's murder
Detectives hunting the killer of a policeman gunned down on Boxing Day today arrested a man in his late 30s.

(1) a man in his late 30s.

Does it mean "a man whose age is about 36 - 39 years old"?

(2) PC: plebeian cop?

(3) Does the sentence "Detectives hunting the killer of a policeman gunned down on Boxing Day today arrested a man in his late 30s." mean "Detectives hunting the killer of a policeman who was gunned down by the killer on Boxing Day today arrested a man in his late 30s."?
That is, "Detectives arrested a man".

I'm almost sure what I thought about (3) is right; still, need you to 100% confirm this.

TIA