All Yordles have something wrong with them mentally
Teemo: Something along the lines of split or multiple personality disorder. Read his lore. Corki: PTSD. He misses war and battle. That's why he joined the league. Tristana: Sadism and megalomania. "Look at the pretty explosions!" Poppy: Seemingly the most sane but she suffers from PTSD (same as Corki) due to witnessing her father's death. Plus there's a possibility of some sort of autism. "I don't have any jokes" or something like that. Lulu : Can't place anything on her. But she lived a long time ago and probably has autism. Also I'm pretty sure Pix is evil and is manipulating her somehow. Veigar: Obvious. SVS (Super Villain Syndrome) and megalomania. Plus he's just downright crazy from his time in prison. Heimerdinger: Can't place anything on him, either, but he talks to himself a lot and is quite narcissistic. His intelligence could be at the cost of Asperger's. Kennen: ADHD. Not mental but still important to note. Also, I've never played Kennen so can someone help me with his quotes so I can place something? Rumble: He has incredibly strong emotions, causing him to be jealous, a megalomaniac and sadistic because he laughs when he torches people. Gnar: He's prehistoric and can turn mega. This is basically split personality, but since we don't know much about him, that's all he has for now. Ziggs: There seems to be a lot of megalomania here. Plus he likes to see explosions and his favourite hobby is demolition. His laughing and carelessness when throwing bombs could also mean sadism. Amumu : Depressed and undead. Also has an anger problem, I mean, his E is "Tantrum"
That's all I could come up with for now. I apologize for any mistakes I might've made with the mental illnesses and/or the quotes and lore, I don 't have a lot of knowledge on the mental illnesses and the symptoms, but I'm still sure that they all have something wrong with them, which is probably why they joined the league.