No, it doesn't.This effect is triggered "When a card is removed from your opponent's security stack," so even when you check multiple cards in your opponent's security stack, it is triggered upon the first check, then you choose whether to activate it or not. Also, this is a [Once Per Turn] effect, so if you activate this effect when a card is removed from their security stack upon the first check, you can't activate the effect again upon the next check.

Yes. Other than [Sea Animal], all traits with [Avian], [Bird], [Beast], [Animal] or [Sovereign] are applicable regardless of other words or pluralizations.

It means that you can use this card's [On Deletion] effect to play "1 red Digimon with [Avian], [Bird], [Beast], [Animal] or [Sovereign] in one of its traits (other than [Sea Animal]), and 3000 DP or less," and if you have a red Tamer when this effect is activated, you add 2000 to the maximum DP for the "3000 DP or less" restriction.For example, you can play 1 Digimon card with 5000 DP or less if you have 1 red Tamer, and you can play 1 Digimon card with 7000 DP or less if you have 2 red Tamers.

No, you can't.It only affects the [On Deletion] effect on this card itself.