Yes. Not seeing the source of your fear means you don't roll attack and ability checks at disadvantage
The frightened condition has one effect that depends on line of sight:
A frightened creature has disadvantage on Ability Checks and Attack rolls while the source of its fear is within line of sight.
The official Sage Advice Compendium pdf offers some guidance on the meaning of "line of sight":
Question: The frightened condition says “while the source of its
fear is within line of sight.” Does that mean you have
disadvantage on attack rolls and ability checks even
if the source is invisible but you have a clear line to its
Answer: No. If you can't see something, it’s not within your
line of sight. Speaking of "line of sight", the game uses the
English meaning of the term, which has no special meaning in the rules.
The Sage Advise Compendium is clear: if you can't see the source of your fear, it is not in your line of sight, and you do not roll at disadvantage.
Let's look back at the three scenarios you offer:
Scenario 1: 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?
Assuming you play with the optional facing rules, a creature "can't see into its rear arc", and would therefore not have a line of sight to the source of its fear if it is behind it.
Scenario 2: 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?
Yes, unless the source of their fear is within the Blindsight radius.
Scenario 3: 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? Assume they would not provoke OA to do this, although the OA is immaterial to the question.
A frightened creature "can’t willingly move closer to the source of its fear", so they would not be able to run back.
Even if it somehow could run back, they would roll with disadvantage as soon as the source of their fear reenters their line of sight.