What this right here says is you're an idiot.
And what this right here says is you're an asshole. I can be polite as you can, so if you want to sprinkle ad hominem attacks in with actual legitimate attempts at argument, I can do that too.
I just explained what fire is. I even highlighted the fact you can have heat/fuel/oxygen and still lack fire.
Which has nothing to do with fire damage. You can take fire damage from a Heat Metal spell, which lacks fuel and may lack oxygen. Not that all fire damage explicitly melts things either, if you must stick to your analogy. Fireball does, but there's no mention of exactly how other fire effects destroy objects they're used on, and the DMG has rules for catching on fire but not for melting.
1. Nonlethal damage doesn't require a type any more than damage does.
This is just a waste of everybody's time. Who said nonlethal damage
needs a type? I just said it
can have a type. I'll admit that actually doesn't explicitly contradict any claim you had made, but it is pretty close to one of them, and I did at least set forth some evidence that you have yet to refute or even address.
Fact: Energy resistance reduces damage of a particular energy type.
Corrolary: Cold resistance reduces cold damge.
Fact: Cold resistance reduces nonlethal damage related to exposure in cold environments.
Possible conclusion 1: As a special rule, certain types of energy resistance can reduce untyped nonlethal damge related to exposure to certain thematically appropriate environments.
Possible conclusion 2: Nonlethal damage related to exposure to cold environments is cold damage.
The passage in Frostburn doesn't specify one way or the other, and the DMG doesn't mention energy resistance, so this is probably up to DM interpretation.
(It's a little surprising that this bit of inductive reasoning continues to elude you, even after I explained it, especially considering the logical leap involved in your induction of the maximum temperature of [Cold] spells. Maybe it really is true that an optimizer's interpretation of the rules is the one that yields the most plusses in the current situation.)
2. RAW the nonlethal damage is not typed as Energy: [Cold] Damage. (I've been using [] markers to denote type or not btw).
Which is not only completely unnecessary, but also wrong. The brackets mark spell descriptors. Damage types like fire and cold are used like normal words in a sentence-- which doesn't exactly help your case that cold environments don't deal cold damage.
If were go with the idea the nonlethal damage from cold is in fact [Cold] Damage then wearing a scarf (or w/e) is capable of preventing the 1d6 cold & 1d4 nonlethal damage of the Unearthly Cold range. But that is in direct contradiction of the rules. Frostburn says protection from cold has nothing to do with protection or resistance to the energy type [Cold]. [Cold] resistance blocks the nonlethal damage as noted (specific) but other effects that blocks the nonlethal damage cannot block [Cold] damage because they were never blocking [Cold] damage to begin with (general).
Now you're the one who needs remedial reading comprehension classes. That passage specifies that the cold protection rules refer specifically to resisting environmental conditions. Notice that someone with cold resistance 5 (or just Endure Elements) just has to put on a cold weather outfit to completely ignore Unearthly Cold (base protection 3 with a +1 modifier). This is in contrast to the general rule that attacks deal damage.
Now you're talking mass and shape dynamics. Those are certainly out of my league and clearly seems out of yours.
I only brought it up because you mentioned the melting temperature of gold, but if you'd prefer to retreat to the simpler realm of game mechanics, I'd be happy to oblige you. (And I did well enough in my materials science classes in university and placed out of all the chemistry I needed, so I doubt it'd be out of my league if I actually did sit down and do some research on it, but that doesn't sound like as much fun as debating the minutiae of D&D rules.)
Speaking of rules, I have a reasonable estimate of how much water an instantaneous [Cold] effect might be able to freeze, given an arbitrarily large body of water.
Since it's not a creature or an effect, water could be modeled using the rules for objects (since just about everything in D&D is either a creature, an object, or an effect). We don't have hardness and HP for water, but we do have it for a reasonable approximation, ice: 0 hardness, 3HP/inch. For large objects (like walls), depleting its HP breaches a 5' section. So, we can consider a 5' square of water to have 180 HP. Normally, cold damage is divided by 4 before applying it to an object, but two rules work in our favor here: Attacks always do at least 1 damage, and the DM may double damage that he feels is particularly effective against an object. Worst case, Ray of Frost deals floor(1d3/4, 1), or 1, damage to a pool of water. Whether you calculate this as a 5'x5'/.3" sheet or 1/180 of a 5' cube is irrelevant; it comes to 0.69(4) cubic feet-- about a 10.6" cube or 6.6" sphere, probably enough for several handfuls and big enough to avoid melting for hours in most climates. Even if the DM decides you only freeze 1/4 of that for doing a partial HP of damage, it's still a minimum of a 6.7" cube or 4" sphere, still a handful for most reasonable hands. This is really good news for our OP, if his DM uses it, since Pathfinder casters get cantrips at will.
But really, the whole thing is in the realm of DM interpretation because there aren't any explicit rules, and I hope for your characters' sake that this isn't how you argue with your DM.