Even though it can generate mana, Dragonrage isn't a mana ability. It uses the stack and can be responded to.

Only creatures that are attacking as Dragonrage resolves will count toward how much mana is generated, and only those creatures will gain the "firebreathing" activated ability. In other words, casting Dragonrage before you've declared attackers usually won't do anything.

Remember that unused mana is lost at the end of each step and phase. For example, if you cast Dragonrage during your declare attackers step, except in very rare cases, the mana Dragonrage generates won't last until the declare blockers step or the postcombat main phase.