The damage is actually lethal regardless of the result of the die.
The rules of combat say:
If your attack succeeds, you deal damage. The type of weapon used determines the amount of damage you deal.
Damage reduces a target's current hit points.
You got 1d4 weapon damage, -2 from strength, +1 from magical enhancement, plus 1d6 fire damage from the Fire enchant.
So your damage roll is 1d4-1+1d6. His damage ranges from 1 to 9.
Meaning that the minimum damage dealt is 1, thus lethal damage.
If penalties reduce the damage result to less than 1, a hit still deals 1 point of nonlethal damage.
See how it doesn't mention weapon damage, but just says damage instead? That means the damage from your attack roll must deal at least 1 point of damage.
Now, pathfinder does separate damage into Weapon Damage and Extra Damage for effects that call for them. Extra damage is never multiplied on a critical hit for instance, one example of that is the extra damage from the Vital Strike feat, which should never be multiplied, but added to the total.
As for weapon damage, the enhancement bonus from being a magical weapon is added on top of your weapon damage. The same goes to your bonus (or penalty) from Strength.
You dont split the die between weapon and extra damage when verifying the rule for minimum damage (because it does not say weapon or extra damage, but simply damage), you check if the total damage was at least 1 or not. If your damage is 0 or negative, you deal 1 point of nonlethal damage. That's it, no further complications.
Say this character takes 6 points of strenght damage, thus got a -3 penalty on all attack rolls and *weapon damage * rolls (which means this penalty only affects damage that is dealt using weapons).
His damage is now 1d4-5+1d6, you could very easily roll 0 or less and still cause lethal fire damage of 1d6. Because his minimum damage is 1, he would still cause lethal damage.
On damage types
If it's relevant, like when a monster has energy resistance or damage reduction, then you need to track the damage type.
Damage Reduction: Some creatures are resistant to certain types of damage. This can be seen in their stats sheet as DR, zombies have their DR as 5/Slashing, while, skeletons got their DR as 5/Bludgeoning.
Feats: Some feats require certain weapon types to be used, like Bleeding Critical requires a slashing or piercing weapon, while Bludgeoner requires a bludgeoning weapon.
Magical Enchantments: Certain magical weapon effects can only be applied to certain weapons types. For example Keen can only be applied to piercing or slashing melee weapons.
Energy Resistance: Energy resistance works just like damage reduction, but instead of weapon damage type, it works on energy damage types, which the five most common are fire, cold, eletricity, acid and sonic. Example, a Shambling Mound will resist the first 10 points of fire damage, regardless of the source.
Your dagger slashing weapon damage would be 1d4-1, and your fire extra damage would be 1d6. If the target has 5/bludgeoning, he would reduce your 1d4-1 damage to 0. Damage Reduction ignores that rule of minimum 1 damage, because the reduction is applied after the damage was calculated.
Or if the target had energy resistance (fire) 10, he would ignore your 1d6 fire damage.
This can also be seen on this answer, here on the site.
Back to our example of a character with 1d4-5 plus 1d6 fire. If this character attacked a creature with 10 fire resistance, his fire damage would be completely negated and that part of his damage roll would cause 0 damage. Because of his 1d4-5 weapon damage, this character would still cause a minimum of 1 point of non-lethal damage.
On sneak attack
The sneak attack damage is actually added on top of the weapon damage (as precision damage type), after the weapon damage has been calculated.
So if you sneak attack with a dagger, you deal slashing sneak attack damage, if you hit with a piercing weapon, its piercing sneak attack damage, and so on.
And it does matter if you deal lethal or nonlethal damage, as all your sneak attack damage will be nonlethal if used with a nonlethal weapon.