Q1: I have 1 card in my security stack. My opponent attacks me with a Digimon with Security Attack +1, and flips over this card during the security check. Its <Recovery +1 (Deck)> activates, adding a card to my empty security stack. Does my opponent continue the security check?
Yes, they continue by checking the card that was added to your security stack.
Holy Wave BT1-107 C [Main] Trigger <Recovery +1 (Deck)>. (Place the top card of your deck on top of your security stack.) --- [Security] Activate this card's [Main] effect.
Holy Wave BT1-107 C [Main] Trigger <Recovery +1 (Deck)> (Take the top card of your deck and place it on top of your security stack). --- [Security] Activate this card's [Main] effect.
Holy Wave BT1-107 C [Main] <Recovery +1 (Deck)>. ---
改心の波動 BT1-107 C 【メイン】≪リカバリー+1《デッキ》≫(自分のデッキの上からカードを1枚セキュリティの上に置く) --- 【セキュリティ】このカードの【メイン】効果を発揮する。