How the new Conqueror actually works:

https://www.youtube.com/watch?v=jhOUm3y-oMo

From this video, is possible to conclude that:

• Stacks are NOT gained or sustained by staying in combat, but rather hitting enemy champions with Basic Attacks or Spells (which includes spells attached to passives like Brand's), which means that DoT effects like Teemo's E passive will not extend Conqueror's duration;

• Stacks are counted for each champion hit, meaning that hitting 5 champions at the same time with "one" basic attack (implies Splash-on-hit effects like Twitch's Ult and Runaan's) ..... will grant maximum stacks immediately;

• The Healing is not penalized for AoE attacks, probably because it only works against champions, meaning that you cannot just attack minions and other easy access targets for a easy heal;

• The Healing counts the true damage dealt by Conqueror (from my calculations and observations, might be wrong though);