The effects trigger simultaneously, so the player can choose the activation order.

This effect directly performs a battle. If this effect is activated, the chosen Digimon battle, as with the standard rules.

Yes, they can battle. A card that has an "effects don't affect" effect can still be chosen for an effect. In addition, a battle itself is a rule that compares DP, not an effect. Therefore, a Digimon that isn't affected by effects can be chosen, and then it can battle. If it loses the battle, it will be deleted as normal according to the rules.

No, you can't activate it.