I'm going to go through, line by line, and just add my additional thoughts, now that I'm on my PC. Doesn't matter whether I agree or disagree with you, your opinion is valid just the same.
: Janna's motivation is very much to help her believers by any means necessary, and I guess I just interpreted this extremely-altruistic motivation of her to be the "Goodest" of the Good Alignments. However, Lawful Good isn't even necessarily the "most good" Alignment, so in this case, I actually think you're probably right.
: Amumu is a case where we have very little motivation or evidence of a character handling moral dilemmas to base an Alignment off of, so I feel his Alignment will be even more subjective than most. Personally, though, since his objective is to make friends, essentially getting people to like him, he would be more prone to conforming to others' expectations of him (that is, if he could even get to know anyone without passively killing them without meaning to), and would likely go out of his way just to get people to like him, at least in my mind. I agree that there's an argument for Neutral Good, but I would still say he belongs in Lawful Good.
: I am fairly confident Jayce is Lawful Good. Some specific examples from his lore that make me think this; he waited for every other scholar in his class to experiment on the crystal he eventually put in his hammer, before he got to, as per the decree of his higher-ups, when he easily could have taken it for himself; in one of his short stories, he has a genuine moral dilemma over helping a little girl, but ends up doing so as he sees it being the right thing to do as Piltovian citizen.
: I'm not sure you actually looked at his lore. He literally wanders the globe trying to enforce ideals of Empathy and Peace. I could still be mistaken, but it seemed pretty clear cut to me when I was reading it.
: Bard is the Wandering Caretaker. As far as I'm concerned, care-taking is pretty necessarily benevolent. I will say that your point of him potentially killing people is fair, however I don't think killing people precludes you from fitting under a Good Alignment, a'la Miss Fortune or Amumu.
: There is a huge chance I was wrong on this one, since I did them alphabetically, Braum came before I had a better idea of the finer parts of this process. His lore does contain a lot of instances of him going against the societal grain, however you're probably right in that he is much more likely to prefer law and order than he is to reject it.
: Completely agree. I didn't want to leave anybody off the list though, and considering Kassadin is a direct rival to Malzahar, I decided it would be fair to put him in the exact opposite Alignment.
: Another person in the comments already pointed out to me that I was basing Pantheon's Alignment on his previous human form, Atreus. Because of this, his Alignment is almost certainly incorrect. That being said, I'm fairly confident Pantheon still lies on the Lawful side of the spectrum, mostly due to his strict adherence to the Rakkorian code of honor and strong sense of authority. The Good vs Evil axis is definitely open though. Again, just my opinion.
: Teemo's lore describes him as a social outcast who still sticks up for his homeland and protects his community. As a rule, Yordles are social beings, hell solitary confinement is what drove Veigar insane. So Teemo's leanings towards solitude, within the community of Bandle City, definitely veer him far enough away from Lawful to be considered Neutral Good, in my eyes.
: I will admit that Vi definitely seems like she should be Chaotic Evil, and I was doubting myself for not putting her there. The fact that she still operates fully within the law was what tugged me towards Neutral Good, but another comment string on this post has already gone into detail as to why Lawful is not a very good term, as it actually refers towards Order, Respectfulness, and value of Authority. With all this in mind, Vi is almost certainly Chaotic Evil.
: While it is true that Zilean doesn't really do much concrete good in his lore, Motives are much more important than Actions regarding Alignment. Just look at Amumu, despite the fact that he ruins lives and landscapes, the character himself intends none of this, and feels a deep remorse for doing so, as it is uncontrollable. Zilean's whole motivation is to find a way to save his people, he is certainly on the Good side of Alignments as I see it, I could see him being either Neutral Good or Lawful Good though.
: Annie has been brought up a couple times now. She is another example of insufficient evidence. I based her Alignment off of what I thought she would inherit from her parents, who would likely be her primary role models. They were originally Mages who lived in Noxus, and after their dissenting opinions towards the crimes of the Noxian Government were silenced, the led an exodus out of the country to live on their own with a settlement of other political outcasts. The empathy displayed by disdain for governmental atrocities, combined with the altruism of helping others escape the country along with them, leans me towards Good, and their strong belief in their own values and the willingness to strike out on their own led me towards Chaotic.
: I honestly had no idea where to put this guy. He is an abstract artist, who produces art for the benefit of both himself and others, shows disdain and contempt for those he sees as unworthy, and values his own freedom above his need to conform to an alien society, all things that led me to TENTATIVELY pin him as Chaotic Good.
: Miss Fortune has a lot going on, and is surprisingly enough one of the more complex characters in League. I would argue that she lies on the Good side of the spectrum, for the reasons that killing doesn't preclude a character from being Good, and everyone I could find her killing in her lore WAS bad. On the Lawful vs Chaotic side, she has strong arguments both for Chaotic and Lawful in my opinion, which would lead to a Neutral conclusion, but I really don't think that fits her well at all, given how passionate she is.
: I really don't think racism of this light a nature precludes her from being a Good character. I would say that it is telling of her anti-social nature, which basically just feeds into her Chaotic Alignment. As to her being Good vs Evil, again motives matter more, in my opinion. She cares less for the fallout from her actions, which again just describes a Chaotic character, and her one true motive is to better the lot of the Vastaya, which I would describe as Good.
: As for Diana, she gave up her whole life to follow the once abandoned path of the Lunari. While this does pin her as counter-culture, which should be Chaotic, it also grants her a strict code of beliefs and values that I think pull her all the way over to the Lawful side. As far as Good vs Evil, I interpreted her character as caring much less about the morality of her actions as long as they complied with her Lunari code.
: Karthus genuinely believes that the living have something to learn from/gain from becoming undead, although has little empathy for any unwillingness to follow him, which I think pegs him as Neutral quite well. His lore also starts with him as a child being morbidly fascinated with death, and he works his way through a church order committed to serving and burying the dead. Religion in general is a pretty Lawful theme, so I put him here and I think it fits him, but I could be wrong. He also has a strict set of beliefs, and not much of an emphasis on freedom or individuality often seen in Chaotic characters.
: These three necessarily have the same Alignment, regardless of what that Alignment happens to be, based on the fact that they are all dedicated to the Kinkou Order. I would argue that True Neutral fits better than Lawful Neutral, based on this quote from Akali's lore: "Order, chaos, light, darkness -- all things must exist in perfect harmony for such is the way of the universe." The Kinkou order specifically tries to preserve a certain amount of Chaos.
: I disagree with you on Singed. Again, you have every right to your opinion, and I see where you are coming from.
Singed lives by himself, disregards societal norms, and generally sows discord wherever he goes. I want to bring up Riven's lore here, where Singed is asked by the Noxian Government to aid in the invasion of Ionia. He agrees, but ends up killing as many Noxians as he does Ionians in what essentially boiled down to a glorified experiment for him. I would, however, cede the point that there's a valid argument for him being Evil as opposed to Neutral.
: Again, this was a case of me basing Alignments off of something other than a champion's current state. I read Sion's current lore first, and his backstory last, so the original, not-dead Sion was freshest in my mind when I entered him. I'll agree that current Sion is more likely Chaotic than he is Lawful. He even kills his own dudes...
: This is one that I think is open for good debate. Part of the definition for Neutral Evil is villains who work mostly towards their own benefit, without regards to the harm they do to others, but also without an intention necessarily to do harm. Gangplank seems like this kinda guy, where he doesn't care about civilization or authority, but if you stay out of his way he won't actively seek you down unless you've wronged him in the past, unlike say Twitch. This is just my personal take, and I think both Lawful and Chaotic Evil are just as valid given sound reasoning.
Holy shit this has exceeded the maximum character limit for a reply. Guess there will be a brief Part 2!