Techinical questions about dual crossbows

I am running a double crossbow build with the weapon set 1 dedicated to Artillery Ballistas and the set 2 dedicated to Grenades, and I do have a few questions about how bonuses apply and are retained when switching weapons:

https://maxroll.gg/poe2/passive-tree/0qgsd00o

I know Artillery is not affected by weapon damage, only +elemental dmg to attacks and +level to skills, if I buy a crossbow with only these two attributes and use it on set 1, will these two attributes keep affecting my ballistas when I switch back to the Grenade-focused set?

Does +life/mana on enemy killed works on Artillery Ballista?

Also on Ignite, I do have Fiery Death on one of my skills, does it still apply if another skill ignites the enemy first?