Does First Strike apply when two creatures Fight each other?
No. First strike only works in combat, and Fight is not a combat ability. See the Comprehensive Rules:

701.12. Fight
701.12a A spell or ability may instruct a creature to fight another creature or it may instruct two creatures to fight each other. Each of those creatures deals damage equal to its power to the other creature.
701.12b If a creature instructed to fight is no longer on the battlefield or is no longer a creature, no damage is dealt. If a creature is an illegal target for a resolving spell or ability that instructs it to fight, no damage is dealt.
701.12c If a creature fights itself, it deals damage to itself equal to twice its power.
701.12d The damage dealt when a creature fights isn’t combat damage.



702.7. First Strike
702.7a First strike is a static ability that modifies the rules for the combat damage step. (See rule 510, “Combat Damage Step.”)