Fire Strike, obviously. Why do you have any doubts there?
I dont think there is any legendary set that fits you. Just use good legendary or MI pieces for each slot. Items like Shadowflame Mantle and Extreminus are very good.
Well, Solael’s Witchblade is mandatory, as well as Manticore or Rhowan’s Crown, the rest is up to you.
You will get chaos and lightning damage naturally, from Fire Strike upgrades. Also, if you go for Abomination, you’ll get even more chaos damage (but then, you’ll need another class, probably).
Yeah, i fond pyromancer being much better choice. Resistance reduction alone for CoF will boost your DPS more than all Arcanist passives combined, and Possession gives the same tankiness as Maiven’s sphere, but it actually increases your DPS instead of decreasing it.
Do yourself a favor and stick with Pyromancer if you want a Fire Strike gunslinger. Fire/Chaos is the only supported damage type by gear for gunslingers and Pyro specializes in that combo.