The casting permission granted by Rivaz’s third ability doesn’t change when you may cast the spell from your graveyard.

The ability that the spell gains as Rivaz’s last ability resolves continues to apply to the permanent that spell becomes after the spell resolves, even if it stops being a Dragon or stops being a creature.

The ability that the spell gains only applies to it once it’s on the battlefield. If the spell is countered before it resolves, that ability will not exile it.

The last ability triggers when you cast a Dragon creature spell from your graveyard for any reason. It triggers even if you didn’t use mana generated by Rivaz’s activated ability to pay its costs.

Once you begin casting a spell from your graveyard, it immediately moves to the stack. Players can’t take any other actions until you’re done casting the spell.

You can use mana generated by the activated ability to pay an alternative cost (such as an escape cost) or an additional cost to cast a Dragon spell. It’s not limited to paying just that spell’s mana cost.