If the mana ability that caused the granted triggered ability to trigger was activated while casting a spell or activating an ability, the triggered ability will resolve (and +1/+1 counters will be added) before that spell or ability.

If Tyvar leaves the battlefield and returns to the battlefield in the same turn, or if Tyvar leaves and another Tyvar appears, the triggered ability granted by the new one is different than the triggered ability granted by the old one. The one granted by the new Tyvar may trigger for each creature you control, even if the ability granted by the old Tyvar triggered for those creatures that turn. (Note that creatures you control won't have the ability granted by the old one anymore, and if you control multiple Tyvars, you won't have the opportunity to activate mana abilities before the "legend rule" gets you down to one Tyvar.)

The ability that Tyvar grants to each creature you control functions independently for each creature. Specifically, the ability can trigger once each turn for each creature you control, not just once total.