Yes, you can. In such cases, you can't add a card from your security stack to your hand, therefore you only place this card in your security stack.

They become face-up security cards that remain revealed. Other than rules that specify face-up security cards, the rules apply in the same manner as standard security cards.

The check is performed with the card left revealed. Other than rules for cards left revealed, the rules apply in the same manner as standard checks.

Yes, it triggers.

Any face-up cards are placed face down, then you shuffle the cards. After shuffling, those cards are left face-down.