To play dumb means to avoid taking responsibility for something. I prefer the direct approach by saying "that person is avoiding taking any responsibility". As you know British people do not enjoy direct communication whch is why we have these phrases because they are easier to say without appearing accusatory.

Yes, 'play dumb' is used. If you are pretending that you know nothing about a situation you can say that you are 'playing dumb'. It's an American expression, but any English speaker would understand it.