No, a Crit Fail doesn't automatically overrule Advantage.
The Advantage and Disadvantage section of the PHB clearly states:
Use the higher of the two rolls if you have advantage... (pg.173)
Nothing in there says anything that would suggest rolling a 1 would cancel out Advantage so you would still take the higher of the two rolls.
The Rolling 1 or 20 section states:
If the d20 roll for an attack is a 1... (PHB pg.194)
So another way to look at it is that the higher of the two rolls becomes the "new" d20 roll for the attack and, so long as you haven't rolled two 1s, doesn't meet the requirements for a Critical Fail.
This is also backed by Mike Mearls (one of the lead designers, though not rules arbiter like Jeremy Crawford, for DnD) on SageAdvice:
Jerry Hunter
@huntinglabrats
@mikemearls A player had advantage on an attack and rolled the first die a 15 and the second a 20, does he get a crit on the attack?
Mike Mearls
@huntinglabrats Yes - the player uses the better of the two results
I think this is pretty cut and dried with just what you have quoted. If you're a Luck Halfling and you roll a 1, you can re-roll it once and use the number. Then if you just don't like any number you rolled, even the re-roll, you can spend a luck point to re-roll (again). One doesn't replace the other, you simply have both features, and both operate exactly as they written.
First Case: Feature then Feat
So if you use Halfling Luck to reroll a 1, can you then decide to use a luck point and still get to choose any of the dice?
Yes.
Example
Let's say you roll to attack, and roll a natural 1. Your halfling luck kicks in:
When you roll a 1 on an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. (PHB, 28)
You get a, say 5. It still isn't high, so you choose to use your luck feat.
Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20... You choose which of the d20s is used for the attack roll, ability check, or saving throw. (PHB, 167)
Second Case: Feat then Feature
Or if you use a luck point and that dice rolls a 1, do you reroll it with Halfling Luck and then still get to choose any?
Not quite. The roll is whichever you choose when you use in the feat Lucky. So, you could choose the 1, and then get to re-roll, as per RAW, but you'd have to use the number of that Halfling luck reroll not the previous two.
Example
So, in this case you roll a 3 and use Luck Feat to roll again, you roll a 1.
Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20... You choose which of the d20s is used for the attack roll, ability check, or saving throw. (PHB, 167)
You choose the 1, specifically to trigger your halfling ability. So, you've now effectively rolled a 1, and get to reroll it:
When you roll a 1 on an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. (PHB, 28)
Best Answer
Yes, the halfling gets to reroll the 1. (The halfling gets to reroll the 1 even if the 8 didn't miss.)
How abilities that allow rerolls — like the hafling Lucky trait — apply to advantage and disadvantage is not immediately obvious, so it's actually covered explicitly in the rules for advantage and disadvantage, even using Lucky as the example (PHB, p. 173; Player's Basic Rules, p. 57):
So yes, you get to reroll the 1. If both dice came up 1 though, you'd only reroll one of them though, not both.