Q1: When I have no face-up security, this card's At the Start of Your Main Phase effect and the digivolved effect of "BT18-004 Prolomon" were triggered at the same time. In this case, can I put a card in my hand face down under security by the effect of "BT18-004 Prolomon" and then use this card's effect to make it Memory+1?
Yes, it is possible.
Winr BT19-084 U LIBERATOR [Start of Your Main Phase] If you have face-up security cards, gain 1 memory. [Main] By suspending this Tamer, 1 of your Digimon may digivolve into a Digimon card in your face-up security cards. If this effect digivolved, you may place 1 Digimon card with the [Royal Base] trait from your hand face up as the bottom security card. --- [Security] Play this card without paying the cost.
Winr BT19-084 U LIBERATOR [Start of Your Main Phase] If you have face-up security cards, gain 1 memory. [Main] By suspending this Tamer, 1 of your Digimon may digivolve into a Digimon card in your face-up security cards. If this effect digivolved, you may place 1 Digimon card with the [Royal Base] trait from your hand face up as the bottom security card. --- [Security] Play this card without paying the cost.
Winr BT19-084 U LIBERATOR [Start of Your Main Phase] If you have a face-up security card, gain 1 memory. [Main] By suspending this Tamer, 1 of your Digimon may digivolve into a Digimon card in your face-up security cards. If this effect digivolved, you may place 1 Digimon card with the [Royal Base] trait from your hand face-up as your bottom security card. --- [Security] Play this card without paying the cost.
サイキヨ BT19-084 U リベレイター 【自分のメインフェイズ開始時】自分の表向きのセキュリティがあるなら、メモリー+1。 【メイン】このテイマーをレストさせることで、自分のデジモン1体を自分の表向きのセキュリティのデジモンカードに進化させる。この効果で進化したなら、自分の手札から、特徴に「ローヤルベース」を持つデジモンカード1枚をセキュリティの下に表向きで置ける。 --- 【セキュリティ】このカードをコストを支払わずに登場させる。