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)
It adds up to a +0.5 to all rolls
The average roll of a die is 10.5, which is just the average value of the numbers 1-20.
We re-roll any die that comes up as 1, and since we stick with whatever that outcome is, we can just replace the 1 in the list with the average roll again, and recalculate the average, giving us an average roll of 10.975.
Therefore, the halfling luck trait is approximately a +0.5 to all d20 rolls.
This is the distribution, as generated from a quick and dirty matlab script I wrote. As you can expect, it's pretty uniform for all values that are not 1 (note that there's still some variation, because it comes from actual random numbers and not an analytic calculation):
Advantage and Disadvantage
I modified my script to see how it would affect advantage and disadvantage.
For disadvantage:
- Normal average: 7.17
- With Halfling luck:7.80
For Advantage:
- Normal average: 13.82
- With Halfling luck: 14.13
We can see that there's about a +0.6 bonus for disadvantage, whereas there's only a +0.3 bonus for advantage. Of course, this makes sense, because rolling a 1 is more impactful for disadvantage than it is for advantage, as the plots make clear.
Also worth noting is that the fact that you only reroll one die doesn't really seem to change the average value. Again, this makes sense because you only have a 1/400 chance of rolling two ones.
Best Answer
If you roll a 1 you can use the Lucky trait to reroll that die.
If you fail the saving throw (with or without triggering Lucky) you can still only reroll one dice because the first quoted rule applies to this too. This can also trigger Lucky but not if the first roll did - that would be using Lucky twice on the same roll.
For your specific example, the 1 would trigger a reroll for Lucky. As this is free there is no reason not to use it. Irrespective of the reroll the 2 is almost certainly cause a failed save. Based on the reroll, you could then decide if spending Indomitable was a good investment.