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

This effect immediately initiates a battle. If this effect is activated, the chosen Digimon battle according to 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 and the attacking Digimon battles one of my opponent's Digimon, deleting it in battle. Then, if my attacking Digimon's attack target successfully becomes one of my opponent's Digimon and it is also deleted in battle, can I use my Digimon's Piercing effect to perform two checks?

No, you only perform one security check for Piercing. Only one security check can be performed during a single attack, even if Piercing activates multiple times. However, if the number of cards checked is modified by Security Attack, that modified number of cards will be checked during 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 battle one of my opponent's Digimon and delete it. Then, if my attacking Digimon's attack target successfully becomes one of my opponent's Digimon, and that Digimon loses the battle but an effect prevents it from being deleted, can I still activate my Digimon's Piercing effect?

Yes, you can. The Piercing effect triggers and activates if the Digimon deletes one of the opponent's Digimon in battle while attacking, and it is resolved immediately before the end of attack timing. Piercing will still trigger even if another battle occurred during the attack in which the Digimon with Piercing successfully attacked and battled a Digimon.

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 winning Digimon's "when this Digimon wins a battle" effect and the losing 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.

Q10: For this card's On Play When Digivolving When Attacking effect, can I choose not to battle the opponent's Digimon after returning its digivolution cards to the bottom of the deck?

No. You must battle the Digimon whose digivolution cards you returned to the deck with this effect if it's possible.

Q11: Can this card's On Play When Digivolving When Attacking effect return the digivolution cards to the bottom of the deck and battle a different Digimon from the Digimon it suspends?

Yes.