[Security] effects take precedence for activation. Upon a check, a [Security] effect will immediately activate without pending activation. For other triggered effects, the turn player activates their effects first.

The card is placed on the bottom of the cards under the Tamer.

No, you can't.

Only the player of those cards can search/look at them. The opponent player can't search/look at them.

It's placed face-up in the trash.

You place them 1 at a time under the Tamer starting with the top card of your deck.

Yes, you can.