Rulings

If Apex Altisaur fights a creature while either of its abilities is resolving, being dealt damage this way causes its second ability to trigger. It may fight several times in a row this way.
If the target creature is an illegal target when Apex Altisaur's last ability tries to resolve, the ability doesn't resolve. If it's a legal target but Apex Altisaur is no longer on the battlefield when the ability resolves (perhaps because it has already died from picking too many fights) the target creature won't deal or be dealt damage.