Roaring Moon ex

If Pikachu ex has full HP and a Lively Stadium is in play, and Roaring Moon ex uses its "Calamity Storm" attack and discards the stadium, will Pikachu ex's "Resolute Heart" Ability prevent it from being KO'd leaving it with 10 HP remaining?

No, it won't. Calamity Storm does 220 damage and that many damage counters are placed onto Pikachu ex, so it no longer has full HP at this point due to Lively Stadium and the "Full HP condition" is not met in order to trigger Resolute Heart; then the stadium is discarded, returning Pikachu ex's maximum HP to 200, and therefore Pikachu ex is KO'd. However, if Calamity Storm does 230 damage (for example, if Roaring Moon ex has a Vitality Band attached), Resolute Heart would work; the attack places 230 damage onto Pikachu ex, thus meeting the "Full HP condition" in order to trigger Resolute Heart; then the Stadium is discarded, and Pikachu ex survives with 10 HP remaining (190 damage on it).

If Roaring Moon ex's "Calamity Storm" attacks a Pokemon without a Rule Box and chooses to discard the Neutralization Zone stadium, is the damage neutralized before discarding the stadium, or does the damage happen after discarding the stadium?

The *choice* is made to discard the Stadium and damage is calculated, however Neutralization Zone stadium prevents that damage from being done. After that the Stadium is discarded.