The two abilities trigger when the enchanted creature deals damage to an opponent of the creature’s controller, not when it deals damage to an opponent of the Helm’s controller. They trigger from any damage, not just combat damage.

If the enchanted creature is both of the listed colors, it will get both bonuses.

With regards to the two triggered abilities that Helm of the Ghastlord grants to the enchanted creature, the point when damage is actually dealt is the only time it matters what color the creature is. If the creature is blue or black at that time, the appropriate ability or abilities will trigger. They’ll resolve even if the creature changes color or loses its Aura.