To that end, I've prepared the following regarding which non-hero classes are locked to each race and how those classes might be explained within the confines of the existing lore (or with as little deviation as possible).
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Standard Races
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Human
Locked Base Classes: Shaman, Druid
- Shaman:
- Tidesages (Counter Culture)
- Power of my Enemy (Anti-orc counter-culture)
- Totems: Harvest Wicker dolls
- Druid:
- Drust druidism (Kul Tiran Culture)
- Harvest druidism (Gilnean Culture)
- Kal'dorei disciple (Night Elven Connections)
- Forms: As Night Elf; emphasis lion manes.
Dwarf
Locked Base Classes: Druid
- Druid:
- Wildhammer
- Forms: Earthen lion, earthen bear (similar to artifact appearances)
Night Elf
Locked Base Classes: Shaman, Paladin, Warlock
- Warlock:
- Defiance of tradition (Counter Culture) [Risk of satyrism]
- Paladin:
- War Priests of Elune (Cultural drift)
- Night Warriors
- Mount: Moonbound Nightsaber
- Shaman:
- Adopted Culture (Draenei)
- Totems: Statues of Elune
Gnome
Locked Base Classes: Shaman, Druid, Paladin
- Shaman:
- Scientific Curiosity (Bind elementals with technology)
- Totems: Gearspring gadgets
- Druid:
- Tinker
- Forms: Mechs
- Paladin:
- Defiance of tradition (Counter Culture)
- Mount: Sacred Goat
Draenei
Locked Base Classes: Rogue, Druid, Warlock
- Rogue:
- Rangari [Don't start with the hooves, stealth is shadow magic]
- Druid:
- Kal'dorei disciple (Night Elven Connections)
- Forms: Marsuul [Cat], Elekk [Bear], Arcane Construct [Moonkin]
- Warlock:
- Defiance of tradition (Counter Culture) [Risk of Man'ari coloration]
Worgen
Locked Base Classes: Monk, Shaman, Paladin
- Monk:
- Gilnean fight club boxer
- Search for Balance
- Learned from Pandaren (new Tushui NPC shipwrecked in Gilneas bay)
- Shaman:
- Search for Balance
- Professional Illusionist
- Totems: Derby Hats
- Paladin:
- Sister Culture (Stormwindian)
- Search for Balance
- Mount: Running Wild, Twoform Clydesdale
Orc
Locked Base Classes: Priest, Druid, Paladin
- Priest:
- Power of my Enemy (Anti-human counter-culture)
- Shadowmoon Devotee
- Priest of Goldrinn
- Field Medic (like Gnomes)
- Paladin:
- Warpriest of Goldrinn
- Power of my Enemy (Anti-human counter-culture)
- Search to end Fel Corruption
- Mount: Dire Wolf
- Druid:
- Shu'halo disciple (Tauren Connections)
- Blessed of Goldrinn (Wild God connections)
- Forms: Frostwolf (Cat), Battleboar (Bear), Tusked Owlbear (Moonkin)
Forsaken
Locked Base Classes: Shaman, Druid, Paladin
- Shaman:
- Dark Shamanism (Elemental Control by coersive force)
- Totems: Gravestones
- Druid:
- Mushroom on the Log (Drust Druidism)
- Forms: As Tauren, but plagued with exposed sores
- Paladin:
- Mad Devotion (pain causes insanity, race-specific debuff "Agony" increases damage, healing, costs, and wounds by 1% per stack, max 10 stacks, resets with Cannibalize)
- Mount: Skeletal Warhorse
Tauren
Locked Base Classes: Rogue, Mage, Warlock
- Rogue:
- Defiance of Tradition (Grimtotem, Counter Culture) [Don't start with the hooves, stealth is shadow magic]
- War Effort Counterintelligence (like SI:7)
- Mage:
- Defiance of Tradition (Counter Culture, Dalaran training)
- Warlock:
- Defiance of Tradition (Grimtotem, Counter Culture)
Darkspear Troll
Locked Base Classes: Paladin
- Paladin:
- Freethinker (Loa devotee)
- Mount: Holy Raptor
Blood Elf
Locked Base Classes: Shaman, Druid
- Shaman:
- Shu'halo Disciple (Tauren Connections)
- Totems: Hovering crystals
- Druid:
- Defiance of Tradition (Counter Culture, Botanist)
- Shu'halo Disciple (Tauren Connections)
- Forms: As Tauren, no horns. Emphasis on enormous eyebrows. Lynx (Cat).
Goblin
Locked Base Classes: Monk, Druid, Paladin
- Monk:
- Prizefighter
- Learned from Pandaren (new Houjin NPC visiting Kezan)
- Druid:
- Tinker
- Forms: Mechs
- Paladin:
- Paladin for Profit
- The power of Gold (Glistens with the Light)
- Mount: Gilded Trike
Pandaren
Locked Base Classes: Druid, Paladin, Warlock
- Druid:
- Mistweaver (let into the Emerald Dream by Yu'lon)
- Forms: August Tiger (Cat), August Oxen (Bear), August Crane (Moonkin), Sky Serpent (Travel)
- Paladin:
- Brewmaster (Way of the light... brew)
- Mount: Rolling Brewkeg
- Warlock:
- Brewmaster (Drunken nightmares summon demons)
- Summons: Pink Elekk (imp), Alemental (voidwalker), Scrumptuous [Cake] (succubus), Indigestion [Terrible Turnip] (infernal), Keglord (dreadlord), Brewguard [hozen] (felguard), Brewhunter [keg pony] (felhunter)
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Allied Races
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Dark Iron Dwarf
Locked Base Classes: Druid
- Druid:
- Druid of the Flame (Firelands connection)
- Forms: As Night Elf, but flame emphasis
Lightforged Draenei
Locked Base Classes: Monk, Rogue, Druid, Warlock
- Monk:
- Learned before Lightforging
- Learned from Pandaren
- Learned from Draenei Monks
- Rogue:
- Rangari before Lightforging
- Druid:
- Kal'dorei disciple before Lightforging (Night Elven Connections)
- Forms: Lightforged Marsuul (Cat), Lightforged Elekk (Bear), Lightforged Warframe (Moonkin)
- Warlock:
- Enslavement of Demons (The one path is the Light)
- Summons: Lightforged variants of standard Warlock summons
Void Elf
Locked Base Classes: Shaman, Druid, Paladin
- Shaman:
- Dark Shamanism (Elemental Control by coercive force)
- Totems: Mini Ethereal elementals
- Druid:
- Unstable Anatomy (from partial etherealization)
- Defiance of Tradition before transformation (Counter Culture)
- Forms: As Blood Elf, but nebulous
- Paladin:
- Netherlight Champion (Void infusion)
- Mount: Ethereal Voidstrider (bandages around whorling energy)
Kul Tiran Human
Locked Base Classes: Paladin, Warlock
- Paladin:
- Tidesworn Warpriest
- Mount: Tidesblessed Shark
- Warlock:
- Defiance of Tradition (Counter Culture)
Highmountain Tauren
Locked Base Classes: Rogue, Mage, Priest, Paladin, Warlock
- Rogue:
- Bloodtotem Tribe
- Mage:
- Defiance of Tradition (Dalaran Transplant)
- Priest:
- Priest of An'she
- Paladin:
- Sunwalker
- Mount: Spirit of Eche'ro
- Warlock:
- Power of my Enemy (Anti-burning legion counter-culture)
Nightborne
Locked Base Classes: Shaman, Druid, Paladin
- Shaman:
- Vintner
- Totems: Arcwine bottles
- Druid:
- Reconnection to tradition
- Forms: Manasaber (Cat), Galaxy Bear (Bear), Leyline emphasis on other forms
- Paladin:
- Defiance of Legion Occupation (Counter Culture)
- Spellblade (Arcane Paladin)
- Mount: Leyline Spider
Mag'har Orc
Locked Base Classes: Druid, Paladin, Warlock
- Druid:
- Laughing Skull tribe (call of the Goregrond Wilds)
- Forms: Same as Orc Druid would use, Laughing skull mask on the shoulder)
- Paladin:
- Lightbound repentant
- Mount: Lightforged Wolf
- Warlock:
- Enslavement of Demons (The one path is the Light)
- Summons: Lightforged variants of standard Warlock summons
Zandalari Trolls
Locked Base Classes: Warlock
- Warlock:
- Loaspeaker
- Summons: Various loa in place of standard warlock summons. Voidwalker = Krag'wa, Dreadlord = Eraka no Kimbul, Infernal = Akunda, Imp = Jani, Succubus = Pa'ku, Felhunter = Gral, Felguard = Gonk
TL, DR: Skim for the race you wish to discuss, read the bullets, and comment accordingly... the goal being to structure the conversation about new class/race combos