Question: One of the main problems I had to deal with, when I started to meditate and study the Dhamma, was the consequence of giving up attachment to reputation and pride.

Of course this is an extremelly positive thing to do, you simply become stronger when you stop caring about how other people see you, you just care about your actions, about karma, about your thoughts.

The problem is that people think you just don't care about anything else, they think you are not commited to your job anymore, to your appearance, to other people feelings... for instance: If they give you a bad feedback and you don't suffer and don't react to it, they think you are ignoring them.

It looks like we need small portions of drama to live in this society, non attachment to reputation makes you look like an alien or a brain washed person. I don't know how to balance that sometimes, any advise?

Answer: 
Maybe don't try to balance anything, just experience. Wanting things to be a certain way is always dukkha.
 Ideally, we should do what is appropriate to the situation(this is easy for Arahants, hard for beginners). So if they think they are being ignored, maybe use the misunderstanding as an opportunity to tell them about what your practicing. Say something like: "I'm not ignoring you, I'm practicing the Dharma and to stay in line with the Dharma I keep my words few but just right, it's called Right Speech. All Dharma practices aim at making the practitioner quit ignoring what makes themselves and everyone suffer, in order to transcend suffering and be happy. So, now can you start to see that I'm very far from ignoring you?"
The right fix, in my personal opinion, would be to respond as best as you can when people tell you such things. Try to see what it is about your reaction that gets on these people's nerves. It's not the case that you have to suffer when making use of social norms and conventions in order to maintain a dignified conversation. Try to see if you can find a way to appear right and normal to these people, without sacrificing your stability.

May I also suggest that, so long as you're not a monk, you don't want to sacrifice your job and relationships for the sake of not wanting to engage in things you don't like just right now. It's the long term well being that you're better off working on - and getting fired or thought lowly of by your friends & relatives isn't going to be part of that. Surely you can find a way to balance things out rightly.

Best of luck.