Q1: What does "effects don't affect" mean, exactly?

This effect prevents a card from being affected by effects. For example, your Digimon won't suspend if it's chosen for a "suspend 1 of your opponent's Digimon" effect, and its DP won't be reduced by 3000 if it's chosen for a "1 of your opponent's Digimon gets -3000 DP" effect.

Q2: Can a card that has an "effects don't affect" effect be chosen for an effect?

Yes, it can be chosen. For example, a Digimon that isn't affected by effects can be chosen for a "suspend 1 of your opponent's Digimon" effect.

Q3: Can a card that has an "effects don't affect" effect be given an effect?

Yes, it can. It won't be affected by it, but it can be given an effect. However, if an effect such as Security Attack is given to a Digimon that isn't affected by effects, the Digimon won't be considered to have that effect.

Q4: If a card is affected by an effect, then it later gains an "effects don't affect" effect, what happens to the effect that was affecting it?

As soon as it gains the "effects don't affect" effect, it will no longer be affected.

Q5: If a card with an "effects don't affect" effect gains an effect then later loses the "effects don't affect" effect, what happens to the effect that it gained?

It will be affected by the gained effect as soon as it can be affected by effects.

Q6: A card that has an "effects don't affect" effect was given an effect that triggers at a timing such as When Attacking. Will the effect trigger if that card later meets the trigger conditions?

If the Digimon isn't affected by effects upon the trigger timing, the effect won't trigger.

Q7: If I activate this card's All Turns effect when multiple of my specified Digimon would leave the battle area simultaneously, are all of those Digimon prevented from leaving?

Yes, all of those Digimon are prevented from leaving. This card's All Turns effect affects all Digimon without having to choose them.