Rulings
Q1: 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.
Q2: When this card leaves the Battle Area outside of Battle, I tried to use this card's first All Turns effect to make another Digimon disappear, but the other effect did not make that Digimon disappear. In this case, does this card leave the Battle Area?
No. You will leave the Battle Area. The "by letting go" condition means that if you cannot do what it says, you cannot do anything else. In this situation, if you cannot make it disappear, you will leave the Battle Area.
Data Sources
Lv.6 Lilithmon (X Antibody) EX7-061 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] [Lilithmon]: Cost 1 [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than in battle, if [Lilithmon] or [X Antibody] is in this Digimon's digivolution cards, by deleting 1 other Digimon, it doesn't leave. [All Turns] [Once Per Turn] When other Digimon are deleted, if it's your turn, you may play 1 level 4 or lower purple Digimon card from your trash without paying the cost. If it's your opponent's, trash their top security card.
Lv.6 Lilithmon (X Antibody) EX7-061 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] [Lilithmon]: Cost 1 [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than in battle, if [Lilithmon] or [X Antibody] is in this Digimon's digivolution cards, by deleting 1 other Digimon, it doesn't leave. [All Turns] [Once Per Turn] When other Digimon are deleted, if it's your turn, you may play 1 level 4 or lower purple Digimon card from your trash without paying the cost. If it's your opponent's, trash their top security card.
Lv.6 Lilithmon (X Antibody) EX7-061 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] [Lilithmon]: Cost 1 [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than in battle, if [Lilithmon] or [X Antibody] in its digivolution cards, by deleting 1 other Digimon, prevent it from leaving. [All Turns] [Once Per Turn] When another Digimon is deleted, if it's your turn, you may play 1 purple level 4 or lower Digimon card from your trash without paying the cost. If it's your opponent's turn, trash the top card of their security stack.
Lv.6 リリスモンX抗体 EX7-061 SR 究極体 | ウィルス種 | 魔王型/X抗体/七大魔王 「リリスモン」:コスト1 【お互いのターン】[ターンに1回]このデジモンがバトル以外でバトルエリアを離れるとき、このデジモンの進化元に「リリスモン」/「X抗体」があるなら、他のデジモン1体を消滅させることで、離れない。 【お互いのターン】[ターンに1回]他のデジモンが消滅したとき、自分のターンなら、自分のトラッシュから、紫のLv.4以下のデジモンカード1枚をコストを支払わずに登場できる。相手のターンなら、相手のセキュリティを上から1枚破棄する。