Can any of my spellcasters cast a spell and still benefit from my avatar's ability?

Yes! As long as you control the spellcaster, the game sees it as a spell 'you' cast, meaning the player.

Is the damage considered air damage?

Yes.

If I give my Sparkmage a poisonous dagger, will it deal lethal damage with its tap ability?

Yes, because it's dealing the damage directly via it's ability. (Damage from a lightning bolt cast by the Sparkmage would not be lethal.)

If I reduce the threshold required to cast the spell (e.g. Mix Aer), does the threshold still count for the Sparkmage ability?

Yes. The Sparkmage ability considers printed threshold on the spell.

Sparkmage

Printings