Can I react to dazzling aurora trigger with a temporal breach on disarming rake to kill the dazzling aurora? Will this stop the dazzling aurora trigger from happening

Small addition

Yes to the play, no to the stop. You can absolutely react to Aurora's end-of-turn trigger, and Temporal Breach on your own Disarming Rake is a legal way to kill the Aurora — but the Aurora trigger still resolves and plays a unit. The trigger is a finalized chain item in a Closed State, so only Reaction-speed things are legal; Hidden covers that (Temporal Breach hid earlier, so it's playable for [0]). Note it's the replayed Rake doing the killing, not the Breach — Breach banishes a unit, so the Rake re-enters and its "when you play me, you may kill a gear" trigger goes on the chain above Temporal Breach, killing the Aurora first. Killing Aurora doesn't matter, though: effects already finalized on the chain always resolve unless countered, and Aurora's effect doesn't need its source on the board (FAQ 6408). To actually stop it you need something that counters the trigger itself or something that prevents a player from playing units.