Gatomon

Q:Can I activate the "when an effect trashes this card from the security stack" effect when this card is revealed from the security stack or when searching/looking at this card from the security stack?

A:No, you can't. You can only activate the "when an effect trashes this card from the security stack" when this card is directly trashed from the security stack.

Q:Can I activate this card's "when an effect removes cards from your security stack" effect when another effect plays this card from the security stack?

A:Yes, you can. If this card is directly played from the security stack, it will be played at the same time it's removed from the security stack, therefore the "when an effect removes cards from your security stack" effect will trigger.