The only role that can viably buy an early Vampiric Scepter is a jungler. This is because lifesteal procs off auto attacks, and in early game you should not be auto attacking creeps because it will push your lane too much.
One of the main ideas behind stacking Doran's is that it provides a cheap advantage for early game harrasment. If two champions with similar base stats are harassing each other, the champ with 2 Doran's will have greater sustain in lane and most likely will win the harrasments. Having more health allows you to win the skirmishes against your opponent, survive jungler ganks, last hit easier, and stay in the lane after a failed jungle gank or harassment.
This does put you behind for getting your first big purchase item, however given that a B. F. Sword (the most common big purchase for AD champs) is 1550g, spending an early 475g on a second Doran's, and consequently getting greater CS (note that after ~25 CS the dorans has paid for itself) and champ kills, can actually allow you to get that B. F. Sword sooner.
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
It adds a percentage of your total damage to your attacks. So yes it increases with your level. Since doran's blade adds "flat" damage it's obvious doran's blade makes the ability stronger. Any item that adds attack damage will increases the power of the passive.
However, early on this is much more effective. Let's say you do 100 damage, a 50% bonus would mean 50 extra damage. Getting 50 extra base damage would give you 75 extra damage, a 150% increase. Adding another 50 damage would give you 100 extra damage, an increase of 133% extra damage.