@Pari1 - While it is clear that you believe Airbnb is in the wrong for trying to charge you for the damages you caused in the host's home, either because they have decided that the damage amount was more than you agreed to pay or because they didn't ask your permission, I'm sorry that you think this has legal merit or constitutes fraud. It doesn't look that way to me.
Based on what you've said above, it sounds to me like Airbnb did contact you several times.
1. when the claim was made because you responded to it,
2. when they thanked your for your response to the claim,
3. when the communication was made that "AirBnB was supposed to reply on how to proceed making the payment", and then
4. multiple emails after the attempt to charge your card.
Of those, the first three were all before trying to collect damage-payment from you. Perhaps you misinterpreted the emails?
The way I see it, by agreeing to the terms and conditions of using the site, you did agree to let them charge you for damages... "Airbnb also reserves the right to otherwise collect payment from you and pursue any remedies available to Airbnb in this regard in situations in which you are responsible for a Damage Claim".
I'm sorry that you feel wronged and wish you well in sorting this out. Airbnb will likely miss you as a user since I'm sure you won't ever use them again after they've done this to you.