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.
No
You run into several problems trying to cast Warding Bond on yourself and have your steed take the damage. You seem to be able to cast warding bond on yourself (which seems fine, you get +1 AC take half damage but then take the damage twice). The target of the spell is the one who gets the AC and the resistance, the caster of the spell is the one who takes the damage. Find steed does not make your horse the caster of the spell therefore you cast it on yourself and your horse and you both get the benefits while you take all the extra damage. Additionally I'm not sure how a horse could wear a ring but (assuming you can get a ring on a horse) you would most likely need three rings (2 for yourself and 1 for the horse).
Best Answer
Rules as Intended: No
Jeremy Crawford has clarified:
So, according to Jeremy's clarification, the damage you receive from warding bond is considered the same source as what damaged the ward. Thus, the only time aura of warding would work on it is if that original damage was caused by a spell.
As always, it is up to the DM to decide how much designer clarifications and intent means at their table.