

Q1: Can I process the part of the effect after "after" in this card's Start of Your Main Phase effect without meeting the "by" condition?
No, you can't. If you don't return 1 TS trait card from your trash to the bottom of the deck, you can't process the part following after "after" in this card's Start of Your Main Phase effect.
Toya Kuga BT26-087 U TS [Start of Your Main Phase] By returning 1 [TS] trait Digimon card from your trash to the bottom of the deck, gain 1 memory. After, you may return 1 [Giant Slayer] from your trash to the hand. [On Play] By trashing 1 [TS] trait card from your hand, <Draw 2>. --- [Security] Play this card without paying the cost.
Toya Kuga BT26-087 U TS [Start of Your Main Phase] By returning 1 [TS] trait Digimon card from your trash to the bottom of the deck, gain 1 memory. After, you may return 1 [Giant Slayer] from your trash to the hand. [On Play] By trashing 1 [TS] trait card from your hand, <Draw 2>. --- [Security] Play this card without paying the cost.
Toya Kuga BT26-087 U TS [Start of Your Main Phase] By returning 1 [TS] trait Digimon card from your trash to the bottom of the deck, gain 1 memory. After, you may return 1 [Giant Slayer] from your trash to the hand. [On Play] By trashing 1 [TS] card from your hand, <Draw 2>. --- [Security] Play this card without paying the cost.
久我橙矢 BT26-087 U TS 【自分のメインフェイズ開始時】自分のトラッシュから、特徴「TS」を持つデジモンカード1枚をデッキの下に戻すことで、メモリー+1。さらに、自分のトラッシュから、「巨神兵器」1枚を手札に戻せる。 【登場時】自分の手札から、特徴「TS」を持つカード1枚を破棄することで、≪2ドロー≫。 --- 【セキュリティ】このカードをコストを支払わず登場させる。