You do not need a new SIM card. You only need the PUK code (Personal Unblocking Key). This is an 8-digit unique code that resets your SIM card and allows you to choose a new PIN.
Yes. Airtel does not charge for providing your PUK code or for unblocking your SIM.
Your SIM becomes permanently unusable. You must visit an Airtel store to get a new SIM with the same number (may involve a small fee and identity verification).