After you suffer 3 or more damage from an attack, recover one shield (up to your shield value).

Does this happen before are after damage has been dealt? I'm assuming after but I'm not 100%.

Yes. You have to take the damage before you recover the shield.

You wouldn't have "suffered" the damage until it has been dealt/dealt with which is going to be after you've lost shields already and took damage cards.

Agreed, otherwise the verbage would look more similar to bossk