Do flaming (and other elemental damage) weapons that have the trip special weapon feature deal their extra damage when they are used in a trip attack and hit successfully?
1 Answer
No, they do not.
The fire (or other elemental) damage dealt by such weapons is extra damage, by definition added to whatever damage the weapon deals under other circumstances. A trip maneuver does not deal damage, and thus does not deal the extra damage either.
This is made explicit under Actions in Combat:
options that increase damage don't cause attacks to deal damage if they wouldn't otherwise do so (such as Vital Strike and trip)