Q1: What does "may battle" mean, exactly?

This effect directly performs a battle. If this effect is activated, the chosen Digimon battle, as with the standard rules.

Q2: If 2 Digimon including a Digimon that has an "effects don't affect" effect are chosen for a "may battle" effect, can those Digimon battle?

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.

Q3: During an attack by my Digimon with Piercing, I activate this card's On Play When Digivolving When Attacking effect, the attacking Digimon and an opponent's Digimon battle, and my opponent's Digimon was deleted in battle. Then, if the attack target successfully becomes an opponent's Digimon and it's also deleted in battle, do I use that Digimon's Piercing to perform 2 checks?

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 checks is modified by Security Attack, that number of cards will be checked in a single security check.

Q4: During an attack by my Digimon with Piercing, I use this card's On Play When Digivolving When Attacking effect to have that Digimon and an opponent's Digimon battle, and my opponent's Digimon was deleted in battle. Then, if the attack target successfully becomes an opponent's Digimon, it loses the battle, but an effect prevents it from being deleted, can I still activate that Digimon's Piercing?

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.

Q5: When does "when this Digimon wins a battle" trigger?

It triggers when the battle is won. When the Digimon with this effect wins a battle, the opponent's Digimon that lost the battle is deleted, then the "when this Digimon wins a battle" effect can be activated.

Q6: Does a "when this Digimon wins a battle" effect also trigger when a battle against a Security Digimon is won?

Yes, it triggers.

Q7: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon that lost the battle is deleted. At such times, in what order can players activate the "when this Digimon wins a battle" effect and the effects that trigger upon the losing Digimon being deleted?

They trigger simultaneously, so the turn player can activate their effects first.

Q8: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon lost the battle. At such times, in what order can players activate the "when this Digimon wins a battle" effect and the loser Digimon's effects such as "when this Digimon would be deleted" and "when this Digimon would leave the battle area" effects?

The "when this Digimon would be deleted" and "when this Digimon would leave the battle area" effects can be activated first.

Q9: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon lost the battle. At such times, can the "when this Digimon wins a battle" effect be activated even if an effect prevents the opponent's Digimon from being deleted?

Yes, it can be activated.