I used this card's [End of Attack] effect to declare playing BT19-014 [Shoutmon EX6] from my hand. At such times, can I declare a DigiXros and place this card itself that activated the [End of Attack] effect under BT19-014 [Shoutmon EX6]?
Yes, you can. If you do, this card will be considered removed before its [End of Attack] effect deletes it, therefore this effect will no longer have a deletion target.