No that's incorrect. Use his/he or her/she if the coat appears to be one that would be worn by a male or female. If uncertain, use male pronouns, which are gender neutral in that scenario.
No, usage makes correctness in language, not people trying to invent some weird conlang they wrongly insist is correct English.
All my English teachers from Grades 1-12.
Using "they" as a singular pronoun would bring out the red pen.