Fast/Easy Heroics + Conquest Completion

(Approx 15min, as a Inquisitor Assassin, including every listed mission, + no bonus tasks)

Korriban
• The Hate Machine
• Armed and Dangerous

Dromund Kaas
• Personal Challenge
• Possessed Hunter

Balmorra
• Toxic Bombs
• A Question of Motivation

Tatooine
• Jawa Trade
• Black Box

Alderaan
• Spring Thaw

Hoth
• Pirated Lockbox

(Approx 50min, as a Trooper Tank, including every listed mission, + some bonus tasks)

Tythoon
• Chamber of Speech

Ord Mantell
• Buying Loyalty

Alderaan
• Survivors (Actually became quite annoying the more I played it)
• Special Delivery

Balmorra
• Industrial Sabotage
• Justice for The Lost
• Colicoid Massacre (Might not be that viable if you're really in a rush)
• Local Predators

Coruscant
• Enemies of the Republic
• Republic's Most Wanted

Tatooine
• Pirate Bullies

Belsavis
• Open Communications (Only Easy if you have stealth)
• Jungle Flight

If you are having any problems with soloing these Heroics, just switch your companion to a healer and it should be smooth sailing, yet a bit slower.

After researching and trying to speedrun these quests, the best optimal way is to have a stealth-focused character with a damage-dealing companion.