Does the dwarf favored class bonus and the ticking damage of elemental touch stack?
Add +1/2 to acid and earth spell or spell-like ability damage.
Choose an energy type: acid, cold, electricity, or fire. You gain a melee touch attack causing 1d6 points of damage of that energy type, along with a special effect described below. You also deal energy damage and the related special effect when you attack with your hands using an unarmed strike, a single claw, or a single slam attack. This bonus damage can never apply to multiple weapons.
Acid: Your touch attack causes 1 point of ongoing acid damage per round for 1 round per three caster levels. The target must make a Fortitude save or be sickened for the duration of the ongoing acid damage.
Meaning that at level 6 you touch someone for 1d6+3 and then for the next two rounds do 4 points of damage a round?
No.
You are adding damage to the spell or SLA. This means you only add the damage one time.
In your example, the favored class bonus lets you add +3 damage (only 3, not more than 3) to the entire SLA, but by adding the +3 damage modifier three times, you really are doing +9 damage which is more than the ability allows.
For reference, note the Wizard arcane school (Evocation) and read the Intense Spells ability.
Admittedly, the wording of the favored class bonus does not specify the "once per spell" rule that Intense Spells does, but it does say it adds a specific amount of damage, not a variable amount that scales with the duration of the spell effect, so without specific text to allow you to apply the bonus damage every round, we must fall back on the default position which, by way of precedence, is established by the Wizard ability.
No.
You are adding damage to the spell or SLA. This means you only add the damage one time.
In your example, the favored class bonus lets you add +3 damage (only 3, not more than 3) to the entire SLA, but by adding the +3 damage modifier three times, you really are doing +9 damage which is more than the ability allows.
If the dwarf cast acid fog wouldn't the damage affect everyone in the cloud, thus adding the damage more then once. For that matter, intense spells doesn't stop you from adding the damage to everyone hit by a fireball.
For reference, note the Wizard arcane school (Evocation) and read the Intense Spells ability.
The wording of the FCB lacks 'once per spell' and it lacks 'Whenever you cast' as well, so it seems a leap to suggest it functions the same. Also one is a favored class bonus that you are giving up something every level for, vs a 1st level class power.
Admittedly, the wording of the favored class bonus does not specify the "once per spell" rule that Intense Spells does, but it does say it adds a specific amount of damage, not a variable amount that scales with the duration of the spell effect, so without specific text to allow you to apply the bonus damage every round, we must fall back on the default position which, by way of precedence, is established by the Wizard ability.
Smite evil and Challenge both add you level to damage against a target, after spending a swift action, but I don't think anyone believes that a Cavalier follows the default position of adding you level to damage only works against evil targets.
I'm not saying that it hasn't been ruled that they function alike, but it isn't readily apparent simply from reading the two dissimilar abilities.
Your analogy fails. There doesn't seem to be any ambiguity about the cavalier's Challenge ability so I find no reason to compare it to Smite Evil to apply arbitrary restrictions from the paladin to the cavalier. If I did find an ambiguity in the Challenge ability, I might use Smite Evil to resolve the specific ambiguity but I still wouldn't add other, unambiguous, Smite Evil restrictions to Challenge.
The favored class ability in your OP is a gray enough area that your or your GM can decide to apply it any way you like.
My suggestion is that if everybody makes random rulings in a random fashion at random times, the game rules dissolve into a puddle of chaos faster that your dwarf could dissolve a goblin's face with his acid.
I don't like chaos. Especially in a set of rules. Rules, by definition, are designed to eliminate chaos, not cause it. Based on that, I try to make my rulings very consistent every time. Consistent with themselves and consistent with other rules.
For me, I make rulings in Pathfinder the way judges make rulings in their courtroom - I look for any established precedence and then make a ruling that is consistent with that precedence. In this case, Intense Spells is a very clear precedence that very clearly defines how to add bonus damage to DOT spells, so my ruling will be consistent with it.
You (obviously) can make your own rulings any way you like.