Q1: When my opponent's Digimon attacks and checks my security, I activate this card's effect to gain 1 memory, moving the memory counter to 1 on my side. Does the attack end, even if my opponent's Digimon has checks remaining from Security Attack +1?

No, as long as there are attacks and effects that have triggered but haven't finished resolving yet, the attack continues regardless of the memory gauge's current position.

Q2: While this Tamer is in my battle area, my opponent's Digimon checks my security stack and flips over BT1-107 Holy Wave. Its Security effect activates, adding a card to my security stack. The number of cards in my security stack remained the same, but can I still suspend this card to gain 1 memory?

Yes, because a card was removed from your security stack.

Q3: While this Tamer is in my battle area, I use BT1-087 Matt Ishida's effect to add a yellow card from my security stack to my hand, triggering <Recovery +1 (Deck)> and adding a card to my security stack. The number of cards in my security stack remained the same, but can I still suspend this card to gain 1 memory?

Yes, because a card was removed from your security stack.

Q4: If my opponent uses a card like BT2-020 Gallantmon to trash a card in my security directly, can I activate this card's effect?

Yes, you can.

Q5: If I activate an effect like BT4-104 Blinding Ray to remove cards from my own security stack, can I activate this card's effect?

Yes, you can.

Q6: If this card is played by its Security effect, can I suspend it to gain 1 memory?

"No, you can't. Since this card hasn't been played when it is revealed from your security, the condition for When a card is removed from your security stack can't be triggered, so this card's effect can't be activated."