[RPG] Does the caster’s resistance, vulnerability or immunity affect damage from Warding Bond

The short answer is that your Order of Operations is correct for both the attack and the Fireball. The rules use the phrasing "takes damage" consistently throughout.

The example given in the Damage Resistance and Vulnerability section (page 197 of the PHB), answers a lot of your questions.

For example, a creature has resistance to bludgeoning damage and is hit by an attack that deals 25 bludgeoning damage. The creature is also within a magical aura that reduces all damage by 5. The 25 damage is first reduced by 5 and then halved, so the creature takes 10 damage.

So, the raw damage is phrased as "an attack that deals 25 bludgeoning damage". Then all the calculations are done, then the final result is phrased "the creature takes 10 damage". Warding Bond says that

each time it takes damage, you take the same amount of damage.

Therefore, the Cleric will take the actual amount of damage that was applied to the Paladin's hit points after all the resistances and saves were calculated.

Fireball won't change the order of operations, since the Dexterity save determines how much damage the Paladin takes. For proof, we can go to Fireball itself:

A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one.

Or, to put it differently, a target takes half as much damage on a successful save.

Damage is never "untyped":

Different attacks, damaging spells, and other harmful effects deal different types of damage. (p. 75 Basic Rules)

Jeremy Crawford confirmed this:

All damage has a type. #DnD

Since it cannot be "untyped" and there is no basis for assuming the type would change, it is the type(s) of the original damage.

The damage types are Acid, Bludgeoning, Cold, Fire, Force, Lightning, Necrotic, Piercing, Poison, Psychic, Radiant, Slashing, and Thunder. (p. 75 Basic Rules)