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.
If you think about it, it won't work to just say: "Oh, you rolled a natural one? Well, there's definitely no secret door there, then!" The players will quickly figure out that low rolls mean you'll tell them the opposite of the truth, and they'll start doing the opposite of whatever you tell them.
When a DM really wants to give false information based on the die roll, they have to do something more complicated: they ask for the player's modifier, roll for the player behind the DM screen, and then give an answer without ever telling the player what they rolled. Some groups like to do this, but I think it's not as much fun when you don't get to roll for yourself.
Here's what the rules say:
If the total equals or exceeds the DC, the ability check is a success—the creature overcomes the challenge at hand. Otherwise, it’s a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the GM.
"Progress combined with a setback" might mean, for example, that the characters eventually find the door but it takes much longer than they expected. Or it might mean the characters find the door but the ivy turns out to be poison ivy, with some associated penalty. Or it might mean that the characters find the door but all the pounding on the walls as they search draws unwelcome attention.
In many cases, the act of searching might take significant time, so the penalty for a failed check is just that the characters get no reward for the time they invested.
But, if there's no obvious setback to be had, it's probably best to just say "no, you don't find anything" and let the players move on.
Best Answer
The roll represents an attempt to do something; the result of that roll, the result of that attempt. (See PHB p.6, "How to Play.")
As such, you can't choose not to attempt to open the door: you've already made the attempt.
I don't know how to say this without sounding snarky, but I do mean it in a helpful manner: there are plenty of roleplaying games where dice don't decide/influence the outcomes of characters' actions. If you don't want to live with dice deciding whether you're sneaky, maybe a different game is the right way to go?
[Reproduced here because, really, it should have been part of the answer and not a comment all along.]