The Spell Vamp portion of the LoL Wiki and my own experience show that
[Spell Vamp] works off of activated item effects, summoner spells, and abilities, regardless of the type of damage done.
Therefore, items that give proc effects to attacks like Bloodrazor, Wit's End, Malady, Wriggles etc, will NOT grant the user health through spell vamp.
On a side note, Ezreal's Mystic Shot (Q) applies on hit effects, including but not limited to, lifesteal.
Here are the answers to this question(s) one-by-one:
What does [Pix's] damage scale from?
Pix's damage scales with Lulu's level. The formula for Pix's damage is:
Pix fires a barrage of 3 bolts, dealing 3 + (2 × level) magic damage each
Does it scale with AD, AP, etc.?
As you can see from the formula above. it does not scale with Attack damage (AD), Ability Powers (AP) or any other champion stat (Mana, Health, etc.) and is based solely on Lulu's level.
Is Pix's damage Magical or Physical?
Pix deals magic damage exclusively. Therefore his damage is dependent on the users Magic Penetration and the targets Magic Resist stats. There is some speculation about whose Magic Penetration Pix uses when making his attacks from an ally (not Lulu) as noted in this link.
Does Pix attack when the ally uses spells, or just for autoattacks?
Pix's attack is triggered by autoattacks only. Casting spells that refresh the attack animation (Nasus's Siphoning Strike, Blitz's Power Fist, etc) may provide additional attacks from Pix but to maximize the damage you probably want to target a champion with a higher attack speed.
For more information about Lulu and Pix check out her entry on the LoL Wikia.
Best Answer
Banshee's Veil prevents any negative magic effect, including damage. This applies to all spells, not just those cast by champions.
I've tested this with three of Baron Nashor's spells:
Banshee's Veil was activated by each of the above spells.
Trivia: