

Q1: Is this card's first effect always active?
Yes, it is always active. The use cost is constantly fluctuating, and the card is referenced as an Option card with that cost.
Q2: Can I process the part of the effect after "after" in this card's Main effect without meeting the "by" condition?
No, you can't. If you don't place 1 Tamer card with either Dan Yuki or Kanan Yuki in its name as any of your Aegiomon's bottom digivolution card, you can't process the part following after "after" in this card's Main effect.
Q3: If I use this card with BT24-085 Dan Yuki & Kanan Yuki's End of Your Turn effect then place that used Dan Yuki & Kanan Yuki card as my Aegiomon's bottom digivolution card, can I process the part of BT24-085 Dan Yuki & Kanan Yuki's effect that follows after "then"?
Yes, you can. If an effect activates, it is to be fully resolved even if the card that activated the effect is removed from that area during the processing.
Q4: I used this card's Main effect to digivolve into Jupitermon. When processing the part of the effect after "after," can I choose a different Jupitermon than the one that digivolved and place the specified card as its top digivolution card?
Yes, you can.
The Thunder Emperor Awakens BT26-097 R TS Add 1 to this card's use cost for each of your security cards. [Main] By placing 1 of your Tamers with [Dan Yuki] or [Kanan Yuki] in their names as any of your [Aegiomon]'s bottom digivolution card, it may digivolve into [Jupitermon] in the hand or trash, ignoring digivolution requirements and without paying the cost. After, you may place 1 card with [Aegiochusmon] in its name in your trash as any of your [Jupitermon]'s top digivolution card. --- [Security] You may play 1 play cost 5 or lower [TS] trait card from your hand without paying the cost. Then, add this card to the hand.
The Thunder Emperor Awakens BT26-097 R TS Add 1 to this card's use cost for each of your security cards. [Main] By placing 1 of your Tamers with [Dan Yuki] or [Kanan Yuki] in their names as any of your [Aegiomon]'s bottom digivolution card, it may digivolve into [Jupitermon] in the hand or trash, ignoring digivolution requirements and without paying the cost. After, you may place 1 card with [Aegiochusmon] in its name in your trash as any of your [Jupitermon]'s top digivolution card. --- [Security] You may play 1 play cost 5 or lower [TS] trait card from your hand without paying the cost. Then, add this card to the hand.
The Thunder Emperor Awakens BT26-097 R TS Add 1 to this card's use cost for each of your security cards. [Main] By placing 1 of your Tamers with [Dan Yuki] or [Kanan Yuki] in their names as any of your [Aegiomon]'s bottom digivolution card, it may digivolve into [Jupitermon] in the hand or trash, ignoring digivolution requirements and without paying the cost. After, you may place 1 card with [Aegiochusmon] in its name in your trash as any of your [Jupitermon]'s top digivolution card. --- [Security] You may play 1 play cost 5 or lower [TS] trait card from your hand without paying the cost. Then, add this card to the hand.
雷霆の覚醒 BT26-097 R TS 自分のセキュリティ1枚ごとに、このカードの使用コスト+1。 【メイン】名称に「結城ダン」/「結城カナン」を含む自分のテイマー1体を自分の「アイギオモン」の進化元の下に置くことで、そのデジモンを手札/トラッシュの「ユピテルモン」に進化条件を無視してコストを支払わず進化できる。さらに、自分のトラッシュの名称に「アイギオテュースモン」を含むカード1枚を自分の「ユピテルモン」の進化元の上に置ける。 --- 【セキュリティ】自分の手札から、特徴「TS」を持つ登場コスト5以下のカード1枚をコストを支払わず登場できる。その後、このカードを手札に加える。