Q1: Can this card's All Turns effect be activated even if one of the players cannot trash one another's security?
No, it cannot. A "by" condition cannot process only a portion of the contents of that condition.
Lv.6 Lucemon (X Antibody) BT19-043 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] Lv.5 or higher w/[Lucemon] in name: Cost 3 [All Turns] [Once Per Turn] When this Digimon would leave the battle area, if a card with [Lucemon] in its name is in this Digimon's digivolution cards, by trashing both players' top security cards, it doesn't leave. [End of Your Turn] [Once Per Turn] Your opponent may trash their top security card. If this effect didn't trash, <Recovery +1 ≪Deck≫> and delete 1 of their Digimon or Tamers.
Lv.6 Lucemon (X Antibody) BT19-043 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] Lv.5 or higher w/[Lucemon] in name: Cost 3 [All Turns] [Once Per Turn] When this Digimon would leave the battle area, if a card with [Lucemon] in its name is in this Digimon's digivolution cards, by trashing both players' top security cards, it doesn't leave. [End of Your Turn] [Once Per Turn] Your opponent may trash their top security card. If this effect didn't trash, <Recovery +1 ≪Deck≫> and delete 1 of their Digimon or Tamers.
Lv.6 Lucemon (X Antibody) BT19-043 SR Mega | Virus | Demon Lord/X Antibody/Seven Great Demon Lords [Digivolve] Lv.5 or higher w/[Lucemon] in its name: Cost 3 [All Turns] [Once Per Turn] When this Digimon would leave the battle area, if a card with [Lucemon] in its name is in this Digimon's digivolution cards, by trashing both players' top security cards, it doesn't leave. [End of Your Turn] [Once Per Turn] Your opponent may trash their top security card. If this effect didn't trash, <Recovery +1 (Deck)>, and delete 1 of your opponent's Digimon or Tamers.
Lv.6 ルーチェモンX抗体 BT19-043 SR 究極体 | ウィルス種 | 魔王型/X抗体/七大魔王 名称に「ルーチェモン」を含むLv.5以上:コスト3 【お互いのターン】[ターンに1回]このデジモンがバトルエリアを離れるとき、このデジモンの進化元に名称に「ルーチェモン」を含むカードがあるなら、お互いのセキュリティを上から1枚破棄することで、離れない。 【自分のターン終了時】[ターンに1回]相手は自身のセキュリティを上から1枚破棄できる。この効果で破棄しなかったなら、≪リカバリー+1《デッキ》≫し、相手のデジモン/テイマー1体を消滅させる。