Yes, they can battle. A card that has an "effects don't affect" effect can still be chosen for an effect. In addition, a battle itself is a rule that compares DP, not an effect. Therefore, a Digimon that isn't affected by effects can be chosen, and then it can battle. If it loses the battle, it will be deleted as normal according to the rules.

No, you only perform 1 security check for <Piercing>. Only 1 security check can be performed during a single attack, even if <Piercing> activates multiple times. However, if the number of security checks is modified by <Security A.>, that number of cards will be checked in a single security check.

Yes, you can. <Piercing> is an effect that triggers and activates if an opponent's Digimon is deleted in battle during the attack, and it's processed immediately before the end of attack timing. <Piercing> will still trigger if the Digimon with <Piercing> successfully attacks a Digimon, even if another battle occurs during this battle.