Q1: Can I use this card's inherited effect to delete another of my level 3 Digimon cards with the Beast trait and then play that deleted Digimon card from the trash without paying the cost?
Yes, you can.
Q2: I used this card's inherited effect to delete a level 3 Digimon card with an On Deletion effect and the Beast trait, then I played that deleted Digimon card from the trash without paying the cost. Can I activate the deleted Digimon's On Deletion effect at such times?
No, you can't. If a card is removed from the trash before a triggered On Deletion effect activates, that On Deletion effect can't be activated.
Lv.5 Antylamon BT17-049 C Ultimate | Virus | Holy Beast [Digivolve] [Turuiemon]/[Wendigomon]: Cost 3 <Alliance> [When Digivolving] You may play 1 level 3 green or yellow Digimon card from your trash without paying the cost. --- Inherited: [End of Attack] [Once Per Turn] By deleting 1 of your other suspended Digimon, you may play 1 level 3 Digimon card with the [Beast] trait from your trash without paying the cost.
Lv.5 Antylamon BT17-049 C Ultimate | Virus | Holy Beast [Digivolve] [Turuiemon]/[Wendigomon]: Cost 3 <Alliance> [When Digivolving] You may play 1 level 3 green or yellow Digimon card from your trash without paying the cost. --- Inherited: [End of Attack] [Once Per Turn] By deleting 1 of your other suspended Digimon, you may play 1 level 3 Digimon card with the [Beast] trait from your trash without paying the cost.
Lv.5 Antylamon BT17-049 C Ultimate | Virus | Holy Beast [Digivolve] [Turuiemon]/[Wendigomon]: Cost 3 <Alliance> [When Digivolving] You may play 1 level 3 green or yellow Digimon card from your trash without paying the cost. --- Inherited: [End of Attack] [Once Per Turn] By deleting 1 of your other suspended Digimon, you may play 1 level 3 Digimon card with the [Beast] trait from your trash without paying the cost.
Lv.5 アンティラモン BT17-049 C 完全体 | ウィルス種 | 聖獣型 「トゥルイエモン」/「ウェンディモン」:コスト3 ≪連携≫ 【進化時】自分のトラッシュから、緑/黄のLv.3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。 --- Inherited: 【アタック終了時】[ターンに1回]レスト状態の他の自分のデジモン1体を消滅させることで、自分のトラッシュから、特徴に「獣型」を持つLv.3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。