Unknown, but perhaps a literary device.
It's likely that Moffat didn't think this through very well, and there's nothing more to it than that. But let's give him the benefit of the doubt and say that it's not intended to be as literal as it might seem.
Like the War Doctor regenerating because he can be the Doctor again, or the nonsensical mess that is The Angels Take Manhattan, Moffat's stories often operate on emotional rather than rational logic.
If we look at it from that point of view, the dread makes sense. Moffat's entire tenure as Doctor Who's producer has been spent telling a story which I've taken to calling "the Myth of Eleven." The audience (and the Doctor) know that Time of the Doctor is the end of that story. The Mysterious Message has ontological weight, a momentum of metaphor and foreshadowing: it signifies the death of the Doctor both literally (the TARDIS tomb at Trenzalore) and on an emotional, meta-textual level (the end of Matt Smith's era).
The dread is an in-story manifestation of the audience's (presumed) dread at losing Matt Smith and reaching the end of the Myth of Eleven. In that sense, it may be Moffat's slightly more subtle version of Russell T Davies' "I don't want to go home." If this is true, the dread is forgotten once it served its purpose. The story was to be a celebration of Matt Smith, not a funeral dirge.
Tackling it literally: addressing your actual points.
Time of the Doctor is a microcosm of Eleven's run as the Doctor, and of Moffat's quirks as its producer. With that in mind, let's look at how the dread lets us recall previous bits of the series.
The Time Lords probably didn't want just anyone poking at the crack, so it makes sense to broadcast a go-away field. However, they're also insanely clever and they know the Doctor very well. Remember the first time we saw Eleven leave Earth? He gave Amy a great lecture about non-interference and then popped out to help a sad little girl. The Time Lords know that nothing gets the attention of the Doctor like a bunch of scared people, and he runs toward danger compulsively. Really, what better way to get his attention than to terrorise the entire population of the universe?
Clara's fearlessness is a bit harder to explain rationally, but if we go for the emotional logic it becomes clear: her companion role is as the Doctor's governess. She keeps him safe, reminds him of what is right, gives him courage and perspective. She's died a million million times across all of history for him. She goes with him because it's her role to chaperone him, and she is fearless because that, too, is her role.
This is easy to address rationally: nobody wanted to go first. The dread alone was never sufficient to keep anyone from going down, but it added to the tense détente over the planet. Recall Eleven's triumphant filibuster the FIRST time he was confronted with a sky full of his worst enemies: "Do the smart thing; let somebody else try first." Which leads to 4.
Once the Doctor broke the détente, the dread still aided him because it kept everyone cautious. But when we're talking about the mightiest races in the universe, faced with the return of their greatest enemy, even a lot of fear will be easy to overcome in order to avoid that fate. Again, remember the Pandorica: out of fear that the universe would be destroyed, the Doctor's enemies united to face the one thing they had feared above all else--the Doctor.
Or maybe they just switched the thing off.
All this is, of course, assuming that the dread effect remained. It's possible that the dread effect was dropped at some point; if we expand logically from the first point, once the Doctor arrived the Time Lords would no longer need to make everyone afraid: it had gotten his attention and kept the Crack unmolested until he arrived. Now that he was there, in fact it would make sense for them to drop the dread so that he'd be under more pressure to speak his name.