Apart from the obvious listing of 
as a friend on her lore page, she has no problem with turning her opponents into cute fluffy animals before killing them (it's almost as if she prefers it that way). She also doesn't seem to care that pix basically stole her whole life away from her, or that her family is dead. Is it possible she is a sociopath?
When looking at Lulu it's important to consider English fairy tales, particularly those involving the fae realms and things like elves, gnomes, and dwarves. These creatures are not particularly violent or cruel* within the expectations and boundaries of their own society*, but have little understanding or empathy for humans and/or human culture. This isn't because they're evil, it's because they're far removed from humanity as whole.
Lulu is very much the same way. She doesn't understand that to many, her magic isn't a game - she grew up among the fey, where changing forms and wild magic are extremely common, and where things like audio/visual perception and the concept of death don't necessarily mean the same thing they do to the rest of Runeterra.
She's not a sociopath. She's a child who never grew up.
Lulu is not evil. In fact, I would argue she is the most innocent champion in LoL. This is the reason why she doesn't see the negative in Veigar. Pix, however, is another story.
First, look at Lulu's kit:
Basic attack - Required because it's part of the video game. "Then only have Pix commit the attack!" Problem there is Lulu loses Pix's attack when she shields an ally.
Passive - Pix adds his own attack to make sure the enemy gets hurt, in case Lulu goes easy on them.
Q - She's actually playing a game with Pix (intersect the beams of glitter!), but unbeknownst to her, the fae magic she shoots hurts others.
W (on enemy) - If you read Lulu's lore, you would know polymorphing was part of the game she played with the yordle children.
E (on enemy) - Pix is the one dealing damage, not Lulu.
R - More of Lulu playing around with her fae magic.
So it's pretty clear Pix is the one with a dark side, not Lulu. Pix is the true attacker. Lulu even says as one of her attack quotes, "Dust 'em, Pix!"
Pix was also the one who lured Lulu to the Glade. For what reason, it's currently unknown (officially). As stated in the lore, when Lulu first found Pix, he was pretending to be stuck in a bird house. That alone envelopes the event in mystery, and the mystery is what keeps Lulu believing Pix is her best friend who would never do her wrong.
Now, the relationship with Veigar...
Both Lulu and Veigar are Bandle City outcasts. That's already clear to everyone. It was what brought the two together, but what exactly created the attraction to Veigar?
After Lulu had joined the League, in one fateful match, she saw Veigar for the first time and asked her lane partner, Tristana, about him. She shared with Lulu his backstory and it brought upon a newfound feeling within her - a feeling of connection. Never before had she been able to feel she could be close to anyone in particular. Everyone was too plain and boring to her liking. Veigar, she felt, shared the same pain and grief she experienced. Perhaps, even, they were much more alike than she knew. Despite her many failed attempts to establish a healthy friendship with Veigar (a.k.a. "Veiggie-poo"), Lulu sticks as close to him as she can without being blown apart by his vast magical power (yet isn't afraid of going for the occasional glomp), even though he could never find it within himself to kill the only yordle who sees the good in him.
Of course Lulu cares about losing her entire family and past life, but she does her best to block it out of her mind. She wants to play and be happy, not be moping and crying everyday. Remember, she isn't aware Pix was consciously responsible for it. She blames herself for staying in the Glade as long as she did, but of course not knowing prior about the time flow differences.
Lulu is schizophrenic. Veigar is the sociopath with Napoleon complex.
She's not evil, she's just... weird. Unlike say, Veigar
She doesn't really seem psychotic as much as just... Blissfully unaware of extent of her own capabilities. Very childish (innocent) in nature. I don't think she really understands how to act in normal society because she lived in the glade so long.
"She led all the children off to play hide and seek, temporarily changing them into flowers and animals to spice up the game, but their parents didn't appreciate her efforts."
It's not like she was trying to do anything bad, she just wanted to play hide and seek, and what better way than as more natural parts of the environment?
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Well she was essentially raised by the fae
and those guys don't do the whole "Morals" thing very well.
She's kind of like Jinx actually, just doesn't kill people.
Good intentions, bad actions
She's full-on crazy and creepy as hell, maybe unintentionally evil? I think her friendship with Veigar is because they both have some hard feelings towards Bandle
Not really, no. I'd label her Chaotic Neutral. She does some pretty dark shit, but usually doesn't want to actively hurt people.
Not evil per se. Veigar is evil, Lulu is amoral; she really does not consider the consequences of her actions or understand why some are so disturbed by her antics.
nah.. i like to think of her as a sort of peter pan of the league of legends world... seriously... the first thing that comes to mind for my is Lulu playing pattycake with pix or leaping between trees having so much fun that most whould look at her as crasy... i whould however try to see and ask if i could join her! odly enough i don't really think she aged over the years...she may be old but not biologically... or mentally for that matter... but that may just be my own odd version of it since my head is known to scew things but i don't care! lulu is all about fun, games, and other things... to be honest any sense of creepy comes from her in-game model.... i think lulu really looks more like gnar (posture wise) and the way she looks now she looks like her whole back is twisted foreword.....
She isn't evil, she is innocent. By innocent, I don't mean good or kind or whatever. Child-like innocents entails child-like ignorance.
The original Peter Pan wasn't evil but he wasn't what we'd call good either.
Lulu is Chaotic-Good. I don't believe Lulu would ever just attack/harm someone for the joy of seeing them in pain/discomfort, she would zap 'em, bop 'em, polymorph 'em as part of a game or what to her is a game.
As someone else has pointed out, Pix is more of the malevolent one. Lulu seems to be a bit misguided by Pix and she's obviously cray to da cray.
Think of her like this:
Like veigar, she was away from fellow yordles. So she doesn't have the happy-go personality like other yordles have. Whatever she was doing or happen to her in the glade, is the result of her personality. I would think endless play, until she decided to come back to Valoran. Man, imagine Lulu being killed and rebirthed on the Shadow Isles? Her killer sure would get....endless play http://fc08.deviantart.net/fs70/f/2013/003/6/7/animation_lol___lulu_by_dakun87-d5q9sht.jpg
Veigar isn't even technically all that evil if you think about it, putting his claims to be to the side for now. He was locked away and went crazy from the solitary confinement. He want's to kill all the evil people in the world so that the world will be peaceful, so he is kind of confused it seems.
As for Lulu, I don't think Pix knowingly stole her away from her family. They were just playing. Not to mention, everyone thought she was weird already before she went into the forest, why would Lulu miss anyone who just made fun of her cuz she was weird?
Maybe we are pix
This is pretty funny, I laughed.
CupcakeTrap, I would send a MIstery Gift for you if my main account was in your server. Thank you for being a writer of nearly every thing that my University group of League of Legends (aprox. 14 summoners) think about this great and flawed Recton. We have weekly meetings and today we were really considering how could the Lore fanbase deal with the bulldozing of the Institute of War, what defined us being what we were and how we presented the game to each new member. While we still are having headaches about to introduce new players to the game (and story matters a lot for our foundational members) we were considering about parting ways with this awful proposal of the New Narrative Team and move on over fanfictions about the Original True Lore. And, man, I do hate fanfictions. To make me do that they really went the wrong way.
You gotta admit her laugh sounds pretty evil... but she's still adorable 
<3
I wish I'm actually evil... taking over the world with Veigar sounds like fun!
Shipping aside, but Lulu is more of a "chaotic neutral". She has her own perceptions of good an evil, no matter what others say otherwise.
Rito never mentioned she was "good" either, she's just "not evil", that's why she's "neutral". She is "chaotic" as her fairy magic goes to.... places. And out of the ordinary according to most people on Runeterra. Come on, it's already strange enough to turn a GIANT monster into a harmless-looking critter! (she can't be lawful, and is definitely not neutral.)
Maybe the Ghandi and Hiter as friends example could be put into use in this context.
