If Dionus leaves the battlefield while one or more triggered abilities generated by the ability it grants are still on the stack, those abilities will still resolve as normal.

A permanent that enters tapped doesn't "become tapped." As such, if it has the ability granted by Dionus, that ability won't trigger.

Dionus's ability grants a triggered ability to itself as well as other Elves you control.

If you somehow control multiple copies of Dionus, each permanent you control will have that many instances of the granted ability. These abilities are not redundant. For example, if you control two copies of Dionus and one becomes tapped during your turn, each of its granted abilities will trigger. When those abilities resolve, each one will untap that Dionus (if it's tapped at the time) and will put a +1/+1 counter on it regardless of whether it's tapped or untapped.

If Dionus leaves the battlefield and returns to the battlefield in the same turn, or if one Dionus leaves and another Dionus enters, the ability granted by the old one is different from the ability granted by the new one. This means that an Elf that already untapped and got a +1/+1 counter from the previous granted triggered ability could untap and get another +1/+1 counter from the new granted triggered ability if it becomes tapped again later that turn.