Q1: Does this card's All Turns effect also trigger when my other Digimon is deleted in battle?
Yes, it is triggered.
Lv.6 TigerVespamon BT18-056 SR Mega | Virus | Cyborg/X Antibody/Royal Base [Digivolve] Lv.5 w/[Royal Base] trait: Cost 3 <Piercing> <Reboot> [On Play] [When Digivolving] For each of your face-up security cards, suspend 1 of your opponent's Digimon. Then, none of their Digimon can unsuspend until the end of their turn. [All Turns] [Once Per Turn] When other Digimon are deleted in battle, this Digimon may unsuspend. (Rule) Trait: Has [Insectoid] Type.
Lv.6 TigerVespamon BT18-056 SR Mega | Virus | Cyborg/X Antibody/Royal Base [Digivolve] Lv.5 w/[Royal Base] trait: Cost 3 <Piercing> (When this Digimon attacks and deletes an opponent's Digimon, it performs any security checks it normally would.) <Reboot> [On Play] [When Digivolving] For each of your face up security cards, suspend 1 of your opponent's Digimon. Then, none of their Digimon can unsuspend until the end of their turn. [All Turns] [Once Per Turn] When other Digimon are deleted in battle, this Digimon may unsuspend. (Rule) Trait: Has [Insectoid] type.
Lv.6 タイガーヴェスパモン BT18-056 SR 究極体 | ウィルス種 | サイボーグ型/X抗体/ローヤルベース 特徴に「ローヤルベース」を持つLv.5:コスト3 ≪貫通≫≪再起動≫ 【登場時】【進化時】自分の表向きのセキュリティ1枚ごとに、相手のデジモン1体をレストさせる。その後、相手のターン終了まで、相手のデジモン全てはアクティブにならない。 【お互いのターン】[ターンに1回]他のデジモンがバトルで消滅したとき、このデジモンをアクティブにできる。 〈ルール〉特徴:タイプ「昆虫型」を持つ。