[RPG] How does Paladin’s Abjure Enethe ability really work

The Frightened condition, emphasis mine (PHB, pg 290):

Frightened

• A frightened creature has disadvantage on ability checks and attack rolls while the source of its fear is within line of sight.
• The creature can't willingly move closer to the source of its fear.

As you say, Dreadful Aspect specifies that the target is frightened of the Paladin. This is specifying that the Paladin is the "source of fear" from the Frightened condition. I'm not sure why this ability calls it out and the Abjure Enemy and Frightful Presence abilities don't, but the source of fear is clear in all cases.

The target will have disadvantage on attack rolls against everyone while the Paladin remains in sight.

If the source of the Frightened condition isn't within line of sight, you're still Frightened, but you don't suffer some of the effects of being Frightened.

• A frightened creature has disadvantage on ability checks and attack rolls while the source of its fear is within line of sight.
• The creature can't willingly move closer to the source of its fear.

There's nothing here that indicates that breaking line of sight ends the Frightened condition. Specific features that inflict the condition might have clauses to that effect, but in general it's not the case.

To answer your specific examples:

If a creature is frightened of you, and they turn their back against you to hit someone else, would they no longer have disadvantage to attack rolls/ability checks?

No, they would still have disadvantage. Even if they turn their back, you would still be within line of sight, since 5e has no facing rules.

If a creature is frightened of you and they have Blindsight, can they close their eyes to stop seeing you, and hence stop rolling with disadvantage?

No. As above, line of sight doesn't require them to actually look at you.

If a creature has 60 ft of movement and is frightened of you, and there is a corner they can reach in 15 ft that will block you from sight, can they run to that corner to stop being frightened, then run back to attack without disadvantage?

They can run to a corner to block line of sight, and stop getting disadvantage on their rolls, but they are still Frightened, and they can't move back, since the second clause of the condition prevents moving closer to the source of their fear. Note that even if they could move back, as soon as the source of their fear was within line of sight, they would be getting disadvantage again.