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: Can I choose a Digi-Egg card for the card returned to the deck with this card's On Play When Digivolving effect?

Yes. However, if you choose a Digi-Egg card, it is returned to the bottom of the Digi-Egg deck instead.

Q4: Can I only return 4 cards from my opponent's trash to their deck to meet the conditions of this card's On Play When Digivolving effect?

No, you can't. All of the conditions within "by doing X" must be fulfilled. The conditions for this On Play When Digivolving effect require you to return 5 cards from the opponent's trash to the bottom of their deck.

Q5: If an effect that activates another effect one time "for every XX" can activate multiple types of effects multiple times, can I activate the same effect multiple times and/or activate different effects?

Yes, you can.

Q6: If an effect that activates another effect one time "for every XX" can activate multiple types of effects multiple times, how is the effect activation order determined?

An effect to activate is chosen, then the next effect to activate is chosen after the previous effect has been resolved. This is repeated one time for every XX.

Q7: If the number of XX changes during activation of effects by an effect that activates another effect one time "for every XX," does the number of activations also change?

No, it doesn't. The number of XX is referenced at the point in time the effect that activates another effect one time "for every XX" is activated.

Q8: Can this card's On Play When Digivolving effect reference this card's own color as well?

Yes, it can. Additionally, it can also reference the color gained by this card's All Turns effect.

Q9: My Digimon and Tamers' colors currently total four. I activate this card's On Play When Digivolving effect and choose the first effect to have it battle an opponent's Digimon with an On Deletion effect, and that Digimon gets deleted. If I then choose the second effect and return that deleted card to the bottom of the deck, will it prevent that returned card's On Deletion effect from activating?

Yes, it will prevent it from activating. If a card with an effect pending activation leaves the area before the effect is activated, then the effect can't be activated.

Q10: When I activate this card's On Play When Digivolving effect and choose its first effect to have it battle one of my opponent's Digimon, my opponent plays a new Digimon with one of their Digimon's immediate-type effects. After that happens, can I use the chosen first effect to battle the opponent's newly played Digimon?

Yes, you can.

Q11: I have this card, 1 "red and blue Digimon," and 1 "red and yellow Tamer" in the battle area. If I activate this card's On Play When Digivolving effect, how many times can I activate "activate 1 effect below"?

Two times. In this situation, your Digimon and Tamers' colors are red, blue, yellow, and white, so you can activate it twice.