2 official rulings from Bandai, last ruled Nov 1, 2024. Straight answers to how this card actually resolves.

I attempted to delete 1 of my other Digimon using this card's 1st <All Turns> effect when this card would leave the battle area other than in battle, but I couldn't delete that Digimon due to another effect. Does this card leave the battle area at such times?
Yes, it leaves the battle area. If the "by doing X" condition isn't met, the rest of the effect isn't processed. In such cases, this card leaves the battle area if the deletion isn't possible.
Ruled Nov 1, 2024
I tried to delete my opponent's EX7-061 [Lilithmon (X Antibody)] using this card's 1st <All Turns> effect. At such times, if my opponent tries to delete this card using the 1st <All Turns> effect on the chosen EX7-061 [Lilithmon (X Antibody)], does this card’s 1st <All Turns> effect trigger again?
No, it doesn't trigger. As soon as the player chooses to activate an [X Per Turn] effect, it will count toward 1 use of [X Per Turn]. In this case, as soon as you attempt to delete another Digimon using this card's 1st <All Turns> effect, it will count toward [1 Per Turn], and the effect won't trigger again during that turn.
Ruled Nov 1, 2024
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