Q1: With this card's On Play effect, can I place cards under 1 of my Tamers if I didn't add cards with Bagra Army in one of their traits from the trash to my hand?
No you can't. First you must add 1 or 2 cards with Bagra Army in one of their traits from your trash to your hand, and then place 1 or 2 cards with the Bagra Army trait under one of your Tamers. If you don't add cards to your hand, you can't place cards under your Tamer.
Lv.6 Lilithmon BT11-087 R Mega | Virus | Demon Lord/Bagra Army [On Play] Trash the top 4 cards of your deck. Then, add up to 2 cards with [Bagra Army] in one of their traits from your trash to your hand, and place up to 2 Digimon cards with [Bagra Army] in their traits from your trash under 1 of your Tamers.[Opponent's Turn] When your opponent moves a Digimon from their breeding area, by trashing 1 of this Digimon's digivolution cards, that Digimon gains "[When Attacking] Lose 3 memory" for the turn.
Lv.6 Lilithmon BT11-087 R Mega | Virus | Demon Lord/Bagra Army [On Play] Trash the top 4 cards of your deck. Then, add up to 2 cards with [Bagra Army] in one of their traits from your trash to your hand, and place up to 2 Digimon cards with [Bagra Army] in their traits from your trash under 1 of your Tamers. [Opponent's Turn] When your opponent moves a Digimon from their breeding area, by trashing 1 of this Digimon's digivolution cards, that Digimon gains "[When Attacking] Lose 3 memory" for the turn.
Lv.6 Lilithmon BT11-087 R mega | Virus | Demon Lord/Bagra Army [On Play] Trash the top 4 cards of your deck. Then, add up to 2 cards with [Bagra Army] in one of their traits from your trash to your hand, and place up to 2 Digimon cards with the [Bagra Army] trait from your trash under 1 of your Tamers. [Opponent's Turn] When your opponent moves a Digimon from their breeding area, by trashing 1 of this Digimon's digivolution cards, that Digimon gains "[When Attacking] Lose 3 memory" for the turn.
Lv.6 リリスモン BT11-087 R 究極体 | ウィルス種 | 魔王型/バグラ軍 【登場時】自分のデッキの上から4枚破棄する。その後、自分のトラッシュから、特徴に「バグラ軍」を含むカード2枚までを手札に加え、自分のトラッシュから、特徴に「バグラ軍」を持つデジモンカード2枚までを自分のテイマー1体の下に置く。 【相手のターン】相手のデジモンが育成エリアから移動したとき、このデジモンの進化元を、選んで1枚破棄することで、このターンの間、そのデジモンに「【アタック時】メモリー-3。」の効果を与える。