Considering most successful writers will openly admit they draw inspiration from all kinds of sources, I hardly think it's somehow "bad". We are a weird species, we take in information from all over the place, all the time, and then mush it into our heads in bizarre ways, that can lead to inspiration for all kinds of things. To try and censor your source material for said inspiration is, I think:
1. Pointless, your brain is going to do it's own thing, and it's going to become creative based on who knows what kind of stimulus you expose it to.
2. Ultimately harmful. You don't have to worry about where you get your ideas from. If it's even remotely original, or hell even if it's not, so what? Do you know how many versions of Romeo/Juliet there are? Nobody cares if the story is good, that it's one they've already seen in a similar fashion.
I've had inspiration for stories, both written, and roleplaying adventures I've created, from some of the most random sources. I mean, would you consider that a song by Peter Gabriel would spark a story for a Werewolf campaign that involves the end of the world, and the battle to prevent that destruction? Well it did. So trying to say things like "the anime and games I watch is going to negatively impact my inspiration" is just kind of silly.
Don't worry about it, just write what you want to write, and put an original spin on the concept. I'm sure you've heard the cliche of "there are no new stories", so why worry?
Now, the two examples you've given for the "bad" inspiration, aren't exclusive to games and anime, you've just given two examples of a poorly structured character. And that's a common problem for writers, to make their stories believable. There are hours and hours of seminars on how to write your characters well. I'd suggest looking up stuff by Jim Butcher (an author who openly admits he pulls inspiration from all over the place), on how to structure a story in a believable way.
Here are two of the links for one seminar he gave that I think is really good, but he's given others.
and then
and I found the other one that I think might be more useful to you, since you gave examples of characters that didn't seem believable.
**EDIT**
And, I'd like to point out, that there isn't anything wrong with your character presenting traits like the two you mentioned above. If you are going for an "anime" style of writing, then it's perfectly fine. Also there are people who are very emotionally mercurial, and can jump moods incredibly quickly. That is a reality, it's not just anime trope.
And there are people in the world who don't really react when they kill people. So again, it's not without precedent, just maybe not for a protagonist, though you could point to hundreds of action heroes from the 80's, who didn't give two shits about making piles of bodies. So you know, your call.