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.