Misogyny, in any form. Belittling or abusing women is all too common in manga and anime. A lot of fan service, like panty shots and other gratuitous stuff (yes, even shower scenes), falls into this category. I'm not saying that a heroine can't get beaten up like a hero would in the same fight but that it shouldn't be in there for the sake of her getting smacked around. Don't go overboard with it. Rape is not something to take lightly and "no" does not always mean "yes." The large amount of manga depicting non-consensual sex acts is a major reason the police want their availability severely limited and I can't say this is unreasonable. Andrea Dworkin perhaps said it best: "Rape is no excess, no aberration, no accident, no mistake- it embodies sexuality as the culture defines it. As long as these definitions remain intact- that is, as long as men are defined as sexual aggressors and women are defined as passive receptors lacking integrity- men who are exemplars of the norm will rape women."
I feel that the most dangerous concepts in humans may be the concepts of face and machismo. "This is how a man acts. He has to be tough and uncaring and dominate all those around him. He has to constantly prove he's a man or he's nobody" Most of the men who believe that are nobody. Look at the places where this remains a strong integral part of the culture or social structure. Most of them are war-torn and/or in poverty aren't they? This sort of thinking usually equivocates violence with masculinity.

The Offensive Little Brat Character. CRAYON SHINCHAN.DR. SLUMP. CHIBI MARUKOCHAN. The last thing that little kids, especially spoiled Japanese boys, need is more bad role models! Look at them on the streets acting out the shows they watch sometimes. These kind of characters aren't heroes if they never grow out of arrogance and overconfidence and rudeness. (I tend to prefer stories where the brats are left behind in despair as the other more reasonable characters evolve.) I truly believe that this is one of the direct causes of some of the social problems previously mentioned.

Then there are these two which really bug me but all I can do is complain about them:

Characters constantly calling the names of other characters. Character A looks at Character B standing silhouetted against the sun ready to attack - "Character A!" Throughout the ensuing fight Character B constantly calls Character A's name. "I'm gonna git you, Character A!" "Character A, you've had it now!"
In a series you need to reestablish the names of the characters in each episode but that doesn't mean that you need to do it every 3 minutes.

Characters named after sets. Food, gemstones, countries, desserts, flowers, animals, houseplants, foreign currencies, types of bread, etc. It was cute and funny the first couple times but now it's like everyone is doing it. It got old four years ago and they're still pounding on it. (This means you Akahori.) If the characters are from the same family or live in a little village then this is not a problem. In a way it compresses the world down to just what you see on the screen. (There can only be so many desserts to name people after...)

Anime is now exported to the entire world. I want to keep it the best kind of animation out there!