Sigils!
pretty sure accuracy doesnt, but its hard to tell since their boosts arent reflected on your character sheet.
i think the sigil of air was nerfed a while ago, but that could be just me. not sure of weapon swap sigils would cause fire or air to proc.
sigil of fire does approximately 90% of what air would hit for. i tested this with steady weapons on golems and the results seem to hold true.
regarding ur statement about the sigil of battle, are u referring to whether a fire/air proc would put the battle on CD? if so, then yes it will. if u proc fire/air, ur weapon swap sigil or any other CD-based sigil will be put on a 5-second CD. according to the wiki, “all sigils share a cooldown; while it is active, no other sigil with a cooldown can trigger.”
i can’t comment on the sigil of accuracy because like forrae said, it’s difficult to tell.