

Q1: Can I trash just 1 digivolution card for the conditions of this card's All Turns effect?
No, you can't. A "by" condition can't be met if only some of the required actions are performed. The conditions for this All Turns effect require you to trash the specified number of digivolution cards.
Q2: Can I use this card's On Play [When Digivolving effect to give "End of Your Turn Delete 1 of your Digimon" to a Digimon that isn't affected by effects?
You can give that effect to such Digimon, but if the Digimon isn't affected by effects upon the trigger timing, that effect won't trigger.
Q3: If I give "End of Your Turn Delete 1 of your Digimon" to an opponent's Digimon, it activates, and my opponent chooses one of their Digimon that isn't affected by my effects, is that Digimon deleted?
Yes, it's deleted. After giving "End of Your Turn Delete 1 of your Digimon" to an opponent's Digimon, it activates as your opponent's own effect.
Q4: If I use this card's All Turns effect to trash 2 cards including EX10-044 Damemon from this Digimon's digivolution cards, then play the trashed EX10-044 Damemon from the trash, can I activate EX10-044 Damemon's inherited effect?
No, you can't activate it. When a card with an effect that's pending activation leaves its current area while activation is pending, the effect can no longer be activated. In this case, being trashed from digivolution cards will trigger EX10-044 Damemon's inherited effect, but it leaves the trash before activation, therefore it doesn't activate.
Lv.6 Lilithmon EX10-058 R Mega | Virus | Demon Lord/Bagra Army [On Play] [When Digivolving] Until your opponent's turn ends, give 1 of their Digimon or Tamers "[End of Your Turn] Delete 1 of your Digimon." [All Turns] [Once Per Turn] When any of your opponent's Digimon are played or deleted, by trashing any 2 of this Digimon's digivolution cards, you may play 1 level 4 or lower purple Digimon card from your trash without paying the cost. ---
Lv.6 Lilithmon EX10-058 R Mega | Virus | Demon Lord/Bagra Army [On Play] [When Digivolving] Until your opponent's turn ends, give 1 of their Digimon or Tamers "[End of Your Turn] Delete 1 of your Digimon." [All Turns] [Once Per Turn] When any of your opponent's Digimon are played or deleted, by trashing any 2 of this Digimon's digivolution cards, you may play 1 level 4 or lower purple Digimon card from your trash without paying the cost. DigiXros -2: 2 Digimon cards w/[Bagra Army] trait When this would be played, you may place specified cards from your hand/battle area under it. Each one reduces the play cost. --- -
Lv.6 Lilithmon EX10-058 R mega | Virus | Demon Lord/Bagra Army [On Play] [When Digivolving] Until your opponent's turn ends, give 1 of their Digimon or Tamers "[End of Your Turn] Delete 1 of your Digimon." [All Turns] [Once Per Turn] When any of your opponent's Digimon are played or deleted, by trashing any 2 of this Digimon's digivolution cards, you may play 1 level 4 or lower purple Digimon card from your trash without paying the cost. DigiXros -2: 2 Digimon cards w/[Bagra Army] trait
Lv.6 リリスモン EX10-058 R 究極体 | ウィルス種 | 魔王型/バグラ軍 【登場時】【進化時】相手のターン終了まで、相手のデジモン/テイマー1体に「【自分のターン終了時】自分のデジモン1体を消滅させる。」の効果を与える。 【お互いのターン】[ターンに1回]相手のデジモンが登場/消滅したとき、このデジモンの進化元を選んで2枚破棄することで、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。 ---