Is there an exchange or provider that actually charge zero transaction fee?

Just curious...

Even blockchain charges 0,0001

Like I say - you can use Bitcoin-qt and pay 0.00000000BTC
but you might risk a very very long transaction confirmation time.

Well, with bitcoin-qt, you can't really send a zero-fee tx unless it is a high-priority one.
You will need to play with raw tx Cheesy

But, you can create a zero-fee tx very easily with Electrum