Battle styles aren't a singular mechanic, but more of a label to designate cards that are designed around a similar theme: for example, Single Strike cards are slower and focus on large, direct damage, whereas Single Strike cards are trickier, messing around with your opponent's resources and keeping them on their toes
Each trigger whether from a land, artifact, creature or another permanent is placed on the stack individually
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If "Dimensional Warrior" attacks "Cyber Jar", the Monster cards that are summoned in the same Monster zones as where Dimensional Warrior and Cyber Jar used to be will be removed from play with Dimensional Warrior's effect
"Mist Archfiend" was once a pretty reliable way to trigger it after taking down an opponent's monster thanks to its high ATK, but this now remains true only in retro formats like HAT and Edison, as it is too slow to be useful nowadays