If multiple sources deal damage to a creature with an enrage ability at the same time, most likely because multiple creatures blocked that creature, the enrage ability triggers only once.

If lethal damage is dealt to a creature with an enrage ability, that ability triggers. The creature with that enrage ability leaves the battlefield before that ability resolves, so it won't be affected by the resolving ability.

If another creature you control is dealt lethal damage at the same time that Bellowing Aegisaur is dealt damage, the other creature won't be saved by the +1/+1 counter that would have been put on it.