Lupas brain is currently bugged - it increases Rexes current health by 100, but not his maximum - once he takes 100 damage, the benefit is gone forever. Notes: You can also use an NCR Guard Dogs brain for this quest, but Rex will gain no additional perk if you do so. Reward: If you give Rex Violetta's Brain, he'll receive a speed increase. Rex is now recruited as a companion, and you'll have this quest, to take him to Jacobstown, in your log. Just speak to The King about Rex, then Julie Farkas over at the Old Mormon Fort, and then return to The King with the news that Rex can be healed.
Trigger: This quest is acquired as you recruit Rex. There is also an inherent time delay between the triggers - you can not trigger Log 2 until 9 days after Log 1, and you will not hear from Lorenzo or Martimer for 2 days after that. Notes: Ed-E's companion quest require that the robot 'overhear' conversations with certain key NPCs about topics such as the Enclave, Pre-War Technology, etc. Reward: If you send Ed-E to the Brotherhood of Steel, he'll receive an upgrade to his armor (+8DT), if you send him to the Followers, they'll a-charge up his lazors (~+50% damage, exact amount unconfirmed). Trigger: Recruit Ed-E, then speak to Johnson Nash in Primm, either in his shop (after installing a new Sheriff), or at the Vicki and Vance (when you first arrive in town). So, this varies wildly by companion, but I'll try to run down all of them.