No the ranger does not get to add his favored enemy bonus twice on an attack. The damage from an enchantment like flaming is still considered part of the weapon damage.
Flaming: Upon command, a flaming weapon is sheathed in fire that deals an extra 1d6 points of fire damage on a successful hit. The fire does not harm the wielder. The effect remains until another command is given.
The word extra damage is the significant part. It does not say it deals an additional 1d6 of fire damage. If the damage was separate than all other bonuses would also be added. Things like the enchantment bonus of the weapon, feats like weapons specialization, and spells like divine favor would all be added to both the physical damage and the energy damage. Considering flaming is a +1 advantage that is way too strong. As it is you are adding on the average 1d6 points of damage, but that is offset by the fact that energy resistance can reduce that damage.
That is correct. Both the fire damage and the ranger EX damage is a bonus to the weapon damage, not the EX is a bonus to all die rolled.
Favored Enemy says "he gets a +2 bonus on weapon attack and damage rolls".
"damage roll" is the operative term here, and where your confusion lies. How many dice are actually rolled, or how many different parts of that damage calculation involve dice being rolled, is irrelevant, the entire stuff (weapon's base damage, plus numerical bonuses, plus additional damage dice) is the "damage roll".
Apart from Manyshot, which is a special case, a successful attack roll is basically always followed by exactly one damage roll, and every HP damage done as part of the attack is part of the damage roll.