Q1: Can the When Digivolving effect of this card designate either my or opponent's Digimon as the target of rest and active effects?
Yes, it is possible. Since neither of them specifies a player, either Digimon can be specified as the target of the effect.
Q2: Does the All Turns effect of this card trigger when either my or opponent's Digimon is active?
Yes, triggers. Yes, triggers regardless of which player's Digimon is active.
Lv.6 Zephagamon ST18-12 SR Mega | Magic Knight/Vortex Warriors/LIBERATOR <Vortex> (At the end of your turn, this Digimon may attack an opponent's Digimon. With this effect, it can attack the turn it was played.) [When Digivolving] Suspend 1 Digimon. Then, unsuspend 1 Digimon. [All Turns] [Once Per Turn] When any Digimon unsuspend, for the turn, this Digimon isn't affected by the effects of your opponent's Digimon and gets +3000 DP. (Rule) Trait: Has [Bird Dragon] Type.
Lv.6 Zephagamon ST18-12 SR Mega | Data | Magic Knight/Vortex Warriors/LIBERATOR <Vortex> (At the end of your turn, this Digimon may attack an opponent's Digimon. With this effect, it can attack the turn it was played.) [When Digivolving] Suspend 1 Digimon. Then, unsuspend 1 Digimon. [All Turns] [Once Per Turn] When any Digimon unsuspend, for the turn, this Digimon isn't affected by the effects of your opponent's Digimon and gets +3000 DP. (Rule) Trait: Has [Bird Dragon] Type.
Lv.6 Zephagamon ST18-12 SR Mega | Data | Magic Knight/Vortex Warriors/LIBERATOR <Vortex> [When Digivolving] Suspend 1 Digimon. Then, unsuspend 1 Digimon. [All Turns] [Once Per Turn] When any Digimon unsuspend, for the turn, this Digimon isn't affected by the effects of your opponent's Digimon and gets +3000 DP. (Rule) Trait: Has [Bird Dragon] type.
Lv.6 ゼファーガモン ST18-12 SR 究極体 | データ種 | 魔法騎士型/ヴォルテクスウォリアー/リベレイター ≪ヴォルテクス≫(自分のターン終了時、このデジモンで相手のデジモンにアタックできる。この効果では、登場したターンでもアタックできる) 【進化時】デジモン1体をレストさせる。その後、デジモン1体をアクティブにする。 【お互いのターン】[ターンに1回]デジモンがアクティブになったとき、ターン終了まで、このデジモンは相手のデジモンの効果を受けず、DP+3000。 〈ルール〉特徴:タイプ「鳥竜型」を持つ。