Q1: An opponent's Digimon attacks using an effect at the end of their turn. Then, if this card was checked upon that Digimon's attack and this card was played using its Security effect, when do I delete the Familiar Token that was played by this card's On Play When Digivolving effect?
It isn't deleted at this timing, it's deleted at the end of your opponent's next turn.
Lv.4 ShoeShoemon P-165 P Champion | Virus | Puppet/LIBERATOR [Security] At the end of the battle, play this card without paying the cost. [On Play] [When Digivolving] Play 1 [Familiar] Token. (Digimon/Yellow/3000 DP/"[On Deletion] 1 of your opponent's Digimon gets -3000 DP for the turn.") At the end of your opponent's turn, delete that token. --- Inherited: <Barrier> (When this Digimon would be deleted in battle, by trashing your top security card, it isn't deleted.)
Lv.4 ShoeShoemon P-165 P Champion | Virus | Puppet/LIBERATOR [Security] At the end of the battle, play this card without paying the cost. [On Play] [When Digivolving] Play 1 [Familiar] Token. (Digimon/Yellow/3000 DP/"[On Deletion] 1 of your opponent's Digimon gets -3000 DP for the turn.") At the end of your opponent's turn, delete that token. --- Inherited: <Barrier> (When this Digimon would be deleted in battle, by trashing your top security card, it isn't deleted.)
Lv.4 ShoeShoemon P-165 P Champion | Virus | Puppet/LIBERATOR [Security] At the end of the battle, play this card without paying its memory cost. [On Play] [When Digivolving] Play 1 [Familiar] Token (Digimon/Yellow/3000 DP/[On Deletion] 1 of your opponent's Digimon gets -3000 DP for the turn). At the end of your opponent's turn, delete that token. --- Inherited: <Barrier>.
Lv.4 シューシューモン P-165 P 成熟期 | ウィルス種 | パペット型/リベレイター 【セキュリティ】バトル終了時、このカードをコストを支払わずに登場させる。 【登場時】【進化時】「使い魔」(デジモン• 黄• DP3000• 【消滅時】ターン終了まで、相手のデジモン1体をDP-3000。)トークン1体を登場させる。相手のターン終了時、そのトークンを消滅させる。 --- Inherited: ≪防壁≫(このデジモンがバトルで消滅するとき、自分のセキュリティを上から1枚破棄することで、消滅しない)