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.
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.
Best Answer
I believe there are two effects that are being confused here: the frightened condition, and the Fear spell.
The frightened condition, as you quoted, specifically prohibits moving towards the source of the fear. It does not matter where the frightened creature is and whether or not it is blocked by anything else, it cannot move towards the source of its fear.
The Fear spell requires that
If there is nowhere to move, the creature who has been targeted by Fear may attack, but it cannot move closer to the enemy, as it is also subject to the frightened condition.
When they attack, in either case, it will be with disadvantage, unless they are under some effect that will allow them to gain advantage, in which case they will cancel out and the frightened creature will attack as normal.