If you only have one question, you should post in the stickied topic rather than making a new one. But to answer your question:
Skills that modify individual unit stats such as Master of Land/Air/Sea and Speed Demon apply in all events where you have to deploy units ( such as tower fights, Training event bosses, coliseum, etc. ), EXCEPT for battle arena. In battle arena, all skills are disabled.
Skills that modify unit skill effectiveness such as Titan Fist, Firebreather, etc. apply in quests but not in boss battles since bosses cannot be afflicted with said skills.
As an aside, Hunter of Darkness is pretty useless, it only applies to episodic bosses and not Training event bosses.