Supporting gay rights goes hand-in-hand with supporting transgender rights. If you tell a cis man who's married to a trans man "You can't REALLY be gay if you're attracted to someone who has PERIODS", then you don't support gay rights. Telling a gay man that he's not actually gay is the opposite of being supportive.
I've never seen a live-action movie where an animal is hit with a tranquilizer dart. In live-action movies, tranquilizer darts are used only on humans, since animals can't act.