Things that really bug me in mainstream anime and manga and that are important reasons that I will not work on shows if the staff won't change them.

Promotion or continuation of ignorance. Example: Getting wet (by falling into water, being in the rain or whatever) and automatically developing a cold within moments. This may seem silly but it's symptomatic of a larger problem. If storytellers rely on wives' tales and superstitions to tell their stories then are they really saying anything of value? I feel letting such things happen is irresponsible. An extension of this would be a character getting AIDS from a toilet seat or from an infected person sneezing on them. It's the same thing.
This also includes punching people in the stomach knocks them out. This is supposed to be a hit to the solar plexus but most people draw it too low. (Ok, so karate chopping or pistol-whipping them on the back of the neck doesn't work in real life either. That is knocking somebody unconscious by inflicting brain damage...) It's rarely a good idea to depress somebody's respiratory system like that. I'm waiting for someone to let out a massive belch or blow their last meal all over the hero when he tries this.

Violence for the sake of violence, shock for the sake of shock, sex for the sake of sex, etc. I am not opposed to having violence in a story. I don't see it as an automatically bad thing. A brutally violent story can sometimes move more viewers than one told in a whisper. It's just when it's there for no reason other than to excite the lowbrows and get them to buy more copies.
You don't see physical abuse as much as before and there's no reason to see it at all. We used to see parents beating their kids and kids beating on other kids (like Bright smacking Amuro, etc.) and it's important to teach kids (and young people and adults) that this is not acceptable behavior in the civilized world.

Fetishes. I'm not talking about sexual fixations here although they do tend to show up quite often. What is it about immature guys and that thing about (as Steve Bennett put it) "infuriatingly large breasts" and their overly ridiculous jiggling, flapping, undulating, oscillating, swaying, vibrating, fluctuating, or gooshing around? Don't you ever get sick of it? I normally avoid anime and manga that is fixated on this but it's really annoying to see in mainstream stuff. Jokes and fixations like this are solely for just-pubescent boys. I just want to grab the directors and scream "grow up!" very loudly into their ears.
Other fetishes include guns, cars, motorcycles, dogs, cats and whatever else the director and chief animators are obsessed with. This doesn't have to be a problem and it isn't unless it gets in the way of the storytelling. Too many times the creator(s) put their obsessions before the story and it makes the story weaker. A great filmmaker like Miyazaki can have his fetishes peppered around a movie but they don't stand out and ruin the world. Although I respect him more than almost anyone else I've ever worked with, a certain famous director at a studio I work with has this very annoying fetish related to certain animals and it leaks into - nay, contaminates - every single thing he does. This is like a famous painter of beautiful portraits having some obsessing with putting small nose hairs in every painting. You want to say, "Yeah, OK, I see you like nose hair but what does this have to do with the art?" If he is making a statement about the imperfection of the human body or whatever then I can forgive it. If it's, "Well, I like nose hair," and it has no other meaning then I have to question it.
Let me say that I don't want to see anyone stopped from waving their fetish around in front of others who may be interested. I'd much rather see it at Comiket, where other like-minded people will be involved, than in mainstream anime and manga though.

Stereotypical voices. It can be hard to do innovative designs but it seems even harder to get voice talent that can do voices outside the stereotypes. As I've said before, I would rather get a totally new person off the street than use a popular but limited voice actor.

Characters not only narrating their thoughts but their actions. "Now I'm going to open this door..." as he opens the door. This is first year comic book amateur stuff! This also covers having unnecessary conversations out loud when they are clearly internal concepts and should be shown through images rather than dialogue. You can wander down the streets and sit on the trains and hear people having audible conversations with non-existent partners. They must think that they're in a movie or that some invisible audience is watching them and understanding what they are doing. It's creepy.

Incorrect body language. This is one of the things I have to correct most often. Chinese or German or Outer-Space-Alien characters should not have the same body language as Japanese characters. It's terribly annoying to me to watch characters who are supposed to be Americans hold one arm behind their heads when embarrassed. I worked on one show where the main character, a German-Italian boy, was arguing with a Greek man and to end the conversation he bowed politely and stomped off. It was really annoying! It doesn't have to be exact, and it's difficult to know about things that you aren't exposed to but if you're going to do it, do it right!

Stereotypes, particularly racial and sexual ones.Nice as a cheongsam may be to look at, the all-too-common image of the Chinese girl in the "China doresu" carrying a bowl of ramen and yattering in pidgin dialogue is like saying, "Har har har! Look at the funny Chinaman!" It doesn't exactly foster understanding and is also extremely outdated and sexist. It's the same as the Irish Cop or Jewish Banker. Caucasians will be drawn pretty much the same as the other characters except with bigger noses. That's just stupid! Anime shouldn't be about close-minded thinking! It really betrays the ignorance of the creators when you see things like this and weakens their message, if they had one at all.
Likewise the sexual stereotypes only serve as a basis for weaker stories. (I do not include the nurse and waitress and bunnygirl and such in this as they are just costumes and do not automatically reflect a single way of thinking and can be used in unique and stereotype-breaking ways.) I'm also pretty tired of people who can't create new personalities or relationships and have to follow the crowd. It's annoying to be able to tell a character's personality from her hair color or from his height in comparison to the other characters or because she wears glasses.