[RPG] How long does the frightened condition last

The Frightened condition isn't supernatural at all.

"Frightened" means the character is overcome with fear, the way people and animals sometimes are. Spells and magical creatures' abilities can cause it, but so can mundane effects like Intimidating Presence.

As such, I concur with HellSaint's answer. The sensible reading of "You can't willingly move closer to the source of your fear" is that you can't move closer to where you think it is. If you move closer to it because it's hiding or invisible or something, then you're doing it by accident, not willingly.

Answers in order

1. The shaken condition caused by demoralizing a foe via the Intimidate skill (facilitated by the blistering invective spell) is replaced by the frightened condition caused by the bard employing a combination of the bard's supernatural ability dirge of doom and the benefit of the feat Improved Dirge of Doom. At least, the condition is replaced for as long as the supernatural ability dirge of doom continues to affect the foe; the affected foes remain demoralized—that duration continuing alongside yet subsumed by the dirge of doom effect—but not also technically simultaneously shaken as fear conditions escalate.

   That is, the supernatural ability dirge of doom doesn't care how the foe gained the condition shaken nor how long the foe would have remained shaken. The dirge ability only checks that foes are shaken and, if they are, then the dirge ability thereafter follows its own rules—until, of course, it can't any longer, like when an affected creature's out of range.

2. A creature that escapes the range of the dirge of doom ability is no longer affected by the dirge of doom ability. If the duration of the creature's demoralization has yet to expire, the creature will become demoralized upon exiting the area. Time spent fightened by the dirge ability, however, counts toward the demoralize effect's duration.

3. To be affected by the dirge of doom ability, "an enemy must be within 30 feet and able to see and hear the bard’s performance," but "[t]he effect persists for as long as the enemy is within [range] and the bard continues the performance." When the enemy who was affected by the dirge ability isn't any longer within range, the dirge of doom effect ceases for that enemy, and that enemy "cannot be affected by it again for 24 hours."

   Thus a bard that's playing spooky music can't just chase after a dude who's already successfully gone beyond the dirge of doom ability's range so as to have the bard's spooky music scare the dude again, despite that chain of events seeming to mesh seamlessly with the adventures of Scooby-doo.

4. The trait Memorable, in part, says, "Whenever you create a fear or mind-affecting effect that imposes a penalty or bonus with a duration of at least 2 rounds, it lasts 1 additional round after it would normally end." While the frightened condition engendered by the benefit of the feat Improved Dirge of Doom does impose a penalty, the supernatural ability dirge of doom itself does not (cf. the spells bane and stage fright that each possess the descriptor fear and that each cause a creature to suffer a penalty directly).

   With this in mind, this GM would err on the side of caution and grant the bard that possesses the Memorable trait no particular advantage for the trait when he uses the ability dirge of doom. However, this GM admits that this ruling may be controversial (q.v. this 2016 Paizo messageboard thread discussing what seems to be an outdated version of the trait Memoriable), so I suggest asking your own GM for a ruling.