Thank you for all these awesome questions! ((It's really helping me flesh out this FAQ page with a lot of common "edge cases"))
In that instance you'd be allowed to either Boost or Shield at Instant Speed! ((Or even crack a Flash Action and have 2 SYN left over which is something I've seen Playtesters pull when they're rocking highly self optimized endgame MECs))
AoEs only ever roll DMG once - If you caused an elemental detonation, that dice boost applies to the whole AoE! ((It's why elementalists really benefit from AoEs!))
For Burning Players can safely remove it from themselves with SYN or an ally using any variety of Actions. If you take a Once Per Allied Phase Action and have burning that will still proc the DMG though. Likewise burning Tension is another way to safely remove Burning.