Actions that match their character type and age. Nothing makes a character more unlikeable than if they act nothing like they should. For example: Children should be naive and innocent, and anything otherwise just makes them creepy, for example. This aspect can be stretched a bit (Like with Littlefoot, who sometimes appears even wise than his elders), but too much, and the character becomes very odd.
I agree that being out of character in a poorly-written way can really take away from a story, but there are some really good stories where this actually happens, and it can make some strong humor and emotion. I know this one fan fiction where an adult, due to some unfortunate circumstances, had the mentality of a child for the most part, while his young nephew often proved to be much more mature than most his age. This could have ruined the story, but the author balanced the differences. There were times where the experience from age did show in this adult when the need arose, and many moments came about when the nephew acted very immaturely.
It added a lot of humor, as the nephew would constantly get irritated with his uncle’s childlike behavior and hurt him, causing the uncle to exaggerate pain and run around. While he mostly acted like a child despite being old, he was able to be a good uncle when it was necessary. As for the child, he basically had to raise himself, so his excess maturity had an in-story reason as well. So characters acting very contrary to their age can really brighten up a story and actually make it more realistic, funny, and emotionally-gripping if handled well and there are good explanations for why things are as they are. Well-done oddities can bring a lot of life to stories rather than tear them down. Some of my favorite anime has some of the oddest characters I’ve seen. Still, I do agree with you that this kind of thing can ruin some stories, I just feel that some authors can make it work.
What I look for in a character sort of depends on what role the character plays.
In the main character, provided he or she is the protagonist (as some focus on the antagonist instead like the main character in the RPG Maker Sample Game) I look for someone who has at least one prevalent weakness and finds a way to either eliminate or overcome that weakness while still holding on to his or her identity. While a change in general is good, I myself prefer gaining strength than becoming weaker. In tragedies and horror frequently, the hero or heroine will become less in the end, and that’s all well and good for a story. I can appreciate that kind of change when I’m in the mood for it, but otherwise, I like seeing flaws being defeated. So my preference is not so much for a character to change, but to change for the better.
As for antagonists, what I enjoy seeing in them most is changing sides at some point. Be it that they were helped by the “good guy,” saw the error of their ways, hurt someone they cared about and needed to fix things, find more to gain from the other side, etcetera, I just love it when the “bad guy” has a change of heart. I know so many anime characters where this happens, and I love it. Prince Vegeta, Tao Ren, Tokiya Mikagami, paper ninja Konan, and so forth. I am not as fond of temporary alliances, but rather true changes of heart. Villains sacrificing themselves to save the hero/heroines is also a very emotional kind of scene.
As for side characters, I just want them adding some unique flavor that no one else does with at least some unexpectedness in behavior from his or her usual routine. Take Toady from the Gummi Bears. Who would have ever thought he would fight against his evil master for the sake of good? Or Ichy from the Past That Yawns Behind fan fiction where he saves someone rather than eating the dinosaur? I just like to get comfortable with a side character for a while to the point I think I know what he or she would or wouldn’t do, then suddenly, I’m shocked!
So in conclusion, I like strength gained in protagonists, the error to be personally seen in antagonists, and for side characters to walk a road for a long time only to make a sudden turn.