A trip attack is NOT a precision strike by any stretch of the imagination, especially when using a tripping weapon that wraps around a limb.
Then you're obviously not tripping precisely enough. You should take a level in Rogue.
Is it an attack? Is the target denied Dex? Does the target have a discernible anatomy?
You're missing the important question- "Are legs vital organs?"
If you want to do Sneak Attack damage, the rules explicitly state that you must strike a vital organ or similarly vulnerable area. IMHO, "legs"- the target of a trip attack- do not meet that criterion, making them an invalid target for SA damage on a trip.
If you want to include "legs" as vital organs, then you might as well drop the requirement that vital organs be struck.
Experiment:
Make a mock tripping weapon- a weight on the end of a rope or string.
Strike with it at a target, intending to hit that target's legs in such a way as to entangle them. If you can't get a target, use your forearm.
You'll note that the weapon takes a lot of time to wrap around the target, and that by the time the weapon has finished wrapping up its target, its kinetic energy is exhausted- it will not hurt to have a limb struck this way (though you may get a rope burn). The weapon will either wrap a large surface area, or it will wrap over its own length, dampening the blow.
So the weapon damage that occurs on a successful hit is replaced, but the energy damage that occurs on a successful hit is not?
Does this not seem inconsistent?
Not to me.
Most weapons harm their targets by transferring kinetic energy from the attacker to the target by slashing, piercing or bludgeoning- quickly, in a single vector and over a relatively small area.
An attack from a tripping weapon does away with that damage by substituting the ability to tangle the limbs of the target in such a way that the target is immobilized in some way. The attack itself does no damage because the force is distributed over a larger area over multiple vectors and takes more time.
In contrast, energy from an enchanted weapon is not kinetic in nature- it is an energy discharge...fire energy, cold energy, electrical energy, whatever. All that is required to discharge the energy is contact- a successful hit.
Experiment:
Buy a nightstick and an electric cattle prod.
Hit someone with the nightstick. Hit someone with the charged cattle prod.
Touch someone with nightstick. Touch someone with charged cattle prod.
Note results. The person hit or merely touched by the cattle prod will experience pain.
The quote you are missing when you talk about "sheathed in fire" and "does not affect the wielder" is:
A flaming weapon deals an extra 1d6 points of fire damage on a successful hit.
No other time. Only on a successful hit and arguably, only when the damage is extra damage.
This is the explanation of what the weapon does. This is the explanation of the game mechanics effect. This explanation does not include what happens on something were there is no hit or where there is no damage.
For example, setting the sword on a stack of papers. There is no "to hit roll" involved with this action. So, what happens?
Again, this is the rules forum where we often distinguish between RAW (explicit rules) and inference (implicit or possibly even intent rules). You do not have an explicit rule here that states that anything touched by the Flaming weapon takes fire damage.
And btw, I 100% agree with you that this is what should happen, but this is not what definitely and explicitly happens by RAW.
You appear to be confusing the two and letting your plausibility override what is actually written.
As for 3.X, considering the section you quoted from the DMG (p303 Catching on Fire), in which non-instantaneous magical fire is listed as a source that can ignite flammables- do you actually consider the weapon's flame instantaneous? Its no more instantaneous than a torch. If you can light something with a torch, you can light it with a flaming sword.
If you use the word instantaneous to refer to the term for spell duration, then no.
If you use the word instantaneous to refer to how long the flame is held against a creature with a trip (e.g. not continuous), then yes.
Again, it is not explicit, so it is interpreted differently by different people.
But, show me in the rules where the exact word "non-instantaneous" is defined as a game mechanic.
I'm not missing the weapon description quote. What you are missing is that the DMG also states that fire is never non-lethal damage in 3.x. I would take this to mean that a weapon sheathed in fire is capable of dealing fire damage whenever it is in contact with something.
Thus, if you set a flaming sword upon a stack of paper, you simply check this list:
1) Is paper flammable- vulnerable to damage from flame? (check)
2) Is the sword in contact with the paper? (check)
3) Is the sword flaming? (check)
4) Is fire normal damage? (check)
As an object the paper should take the damage from the flame. Because of its nature, it should ignite.
An as you have no doubt noticed, I support the position that a flaming (or similarly enchanted) weapon can deal its enchantment damage upon contact with something vulnerable to that damage type. Physics doesn't care which target hits what- energy gets transferred when 2 objects come into contact. A foe grabbing your flaming sword is no less endangered by it than the foe you strike with it.
I'll reply this way- if you run your combats in such a way that a torch used in combat as an improvised weapon can ignite flammables, then there is no logical or mechanical reason that a flaming sword should be run differently- it is no more "instantaneous" than the mundane weapon; its flame is no less lethal.
If, OTOH, you run your combats in such a way that a torch used in combat cannot ignite flammables, then neither should your flaming weapons.