Pretty sure most signs pointed towards a lot more than just that, though, neo. Lou's home life would probably have ended up with at least her mom in jail, were it in more modern times. Her mother wasn't very good at being one. Heart was arguably in the right place, but that was about it.
Also not just being looked down on, if the work was much to go by. You had folks openly mocking her in the middle of classes -- multiple ones, and by all appearances ones not part of any habitual harassing group. That points to a pretty seriously messed up social situation, to say the least. "Looked down on" doesn't cover the likely severity when stuff like that's happening.
L being abused and suffering an apparently wide spread and persistent bullying etc. campaign... well, it still doesn't excuse being abusive (to the extent it actually was, and not just anime tropes being themselves), because precisely bloody nothing does that, but it certainly explains a lot of it. Significant social maladaption/ostracization, severe class-related issues, all on top of an upbringing that would be abusive by almost any standard and hammered in a really messed up set of values, particularly in relation to personal strength. Then tossed on top of a series of events that would break a lot of (most, really) people in two, psychologically. If you're trying to take the series particularly seriously on that front you're dealing with a spectacularly fucked up set of main characters.
IF that was what they were going for. If they were presenting us with a horribly flawed human being who does terrible things. (Like say... Evangeleon does where the hero is a completely messed up anti-hero who normally would be getting psychiatric help but the situation forces him back in only making him worse and worse and to give the show credit.)
I mean... They do, but the thing to remember about Familiar of Zero is it has, conveniently, zero self awareness.
"Lou has low self-esteem so we should feel sorry for her and laugh when she abuses others. Isn't it funny how she constantly abuses people?... Man it is still funny even when no one is making fun of her! Wow now she is the greatest mage ever and she is STILL abusing people!"
The part where I actually put LoZ into the box of "Terrible forever" wasn't the abuse scene... It was when the main character had a dream he was home again... but cried because he wouldn't be with Lou ever again...
Thanks show... for this lovely story about a boy kidnapped from his home and forced into servitude with a horrible abuser... And we are meant to treat it as a perfect love story... I actually complimented the abuse the first time I saw it because I was naïve enough at the time to believe it was furthering the setting and emphasizing just how much of a slave he is and how commoners are treated (But nope... The scene is meant to be funny and sexy! Wow! Look at how he cries in fear!)
And the thing is... She isn't so damaged she can only really handle people in an abusive way... Why? Because we see another person she has love for and she doesn't treat him anywhere NEAR the same was she treats him.
The anime make excuses... and thinks it gets away with it.
A mother abusing her child, isn't a good mother because she was yelled at, at work. If Familiar of Zero did it... The mother would win mother of the year and the scene would be considered hilarious comedy!
Also you missed the BIGGEST part about Familiar of Zero Frumple.
Lou isn't looked down upon anywhere close to as much as they thinks she is... It is almost all in her head. It is why in spite of the fact that she has a familiar almost everyone secretly fawns over AND is the second strongest familiar that she still thinks she has the worst one.