Rulings
Q1: What happens if this card's On Play When Digivolving effect is given to a Digimon that activated a "this Digimon isn't affected by your opponent’s effects until the end of your opponent's turn" effect?
A Digimon with a "this Digimon isn't affected by your opponent’s effects" effect can also be chosen as a target for this effect, therefore your opponent's Digimon will be affected by this card's effect as soon as it loses the "this Digimon isn't affected by your opponent’s effects" effect. As a result, it's deleted at the End of Your Turn timing.
Q2: What does this card's "when ... would leave the battle area other than by ... a battle" mean, exactly?
"When ... would leave the battle area other than by battle" refers to when this Digimon is placed in the trash other than as a result of a battle, returned to the hand/deck, placed in the security stack, moved to the breeding area, or placed under another card.
Q3: Can I use this card's All Turns effect to delete an opponent's level 5 or lower Digimon and prevent this card from leaving the battle area?
Yes, you can.
Data Sources
Lv.6 Lilithmon EX6-057 SR Mega | Virus | Demon Lord/Seven Great Demon Lords [On Play] [When Digivolving] Until the end of your opponent's turn, 1 of their Digimon gains "[End of Your Turn] Delete this Digimon." [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than by battle, by deleting 1 level 5 or lower Digimon, prevent it from leaving. [Opponent's Turn] [Once Per Turn] When another Digimon is deleted, trash the top card of your opponent's security stack.
Lv.6 Lilithmon EX6-057 SR Mega | Virus | Demon Lord/Seven Great Demon Lords [On Play] [When Digivolving] Until the end of your opponent's turn, 1 of their Digimon gains "[End of Your Turn] Delete this Digimon." [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than by battle, by deleting 1 level 5 or lower Digimon, prevent it from leaving. [Opponent's Turn] [Once Per Turn] When another Digimon is deleted, trash the top card of your opponent's security stack.
Lv.6 Lilithmon EX6-057 SR Mega | Virus | Demon Lord/Seven Great Demon Lords [On Play] [When Digivolving] Until the end of your opponent's turn, 1 of your opponent's Digimon gains "[End of Your Turn] Delete this Digimon." [All Turns] [Once Per Turn] When this Digimon would leave the battle area other than in battle, by deleting 1 level 5 or lower Digimon, prevent it from leaving. [Opponent's Turn] [Once Per Turn] When another Digimon is deleted, trash the top card of your opponent's security stack.
Lv.6 リリスモン EX6-057 SR 究極体 | ウィルス種 | 魔王型/七大魔王 【登場時】【進化時】相手のターン終了まで、相手のデジモン1体に「【自分のターン終了時】このデジモンを消滅させる。」の効果を与える。 【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、Lv.5以下のデジモン1体を消滅させることで、離れない。 【相手のターン】[ターンに1回]他のデジモンが消滅したとき、相手のセキュリティを上から1枚破棄する。