firing vs shooting

firing and shooting both are nouns.

Word	Noun	Adjective	Verb	Adverb
firing	Yes	No	No	No
shooting	Yes	No	No	No

As nouns, shooting is a hypernym of firing; that is, shooting is a word with a broader meaning than firing:

Other hypernyms of firing include shot.

firing (noun)	shooting (noun)
the act of setting something on fire	the act of firing a projectile
the termination of someone's employment (leaving them free to depart)	killing someone by gunfire
the act of discharging a gun
the act of firing weapons or artillery at an enemy