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)
No, the wolf would not understand and would not be frightened.
The ability's description is clear in this regard.
The creature understands you only if the two of you share a language.
Since the halfling and the wolf don't share a language, any of the telepathy would be nonsensical to the wolf -- even if it is just growling in an attempt to frighten them.
Animals aren't intelligent, and the wolf likely wouldn't be able to identify that the halfling was the one doing the growling. All it would know is suddenly it can hear phantom growling in it's head, and might not even recognize this as something worth fearing, let alone specifically being afraid of the halfling.
DM fiat applies, of course. If you think it is fine and have no issue with it happening, then allow it. It's your game.
Best Answer
You do not apply advantage separately to the effects of a single saving throw, but you do not get advantage in this case
Most spells require only a single saving throw. This is explicit from the way the save is described, as in dissonant whispers, which says:
This is singular - there’s only one save, even though there are multiple effects. Spells are explicit about the effects of failing the save; in the case of dissonant whispers:
While many spells (mostly cantrips) leave it at that, which tells us that nothing happens on successful save, dissonant whispers is also explicit about that case:
So both effects are covered by the single saving throw: you either take full damage and flee if able, or take half damage and don’t have to flee. Having advantage or disadvantage, however you get it, would apply to the saving throw as with any other d20 roll - you roll twice and choose the result you want. It does not split the saving throw into separate parts to which advantage or disadvantage may or may not apply.
There are a small number of spells with separate saving throws for separate effects, but where this is the case each saving throw is mentioned explicitly in the spell’s description.
In the specific case of a halfling getting advantage on the save for dissonant whispers, though, you’re out of luck. The halfling trait Brave says:
Frightened here refers to the condition detailed in Appendix A of the Player’s Handbook and Basic Rules. Spells and effects will use phrasing like “on a failed save, the target is frightened” to make it clear when it is being used. Dissonant whispers does not mention being frightened, and so Brave does not apply. (Narratively, the movement is framed as a result of the “discordant melody” causing “terrible pain”, not fear.)
Of note is that no spell combines damage with inflicting the frightened condition in a single effect. Effects that cause a target to become frightened may specify additional effects that apply while the target is frightened (for example fear, which forces a character to flee each turn they remain frightened, or weird, which inflicts a separate save against damage on the frightened target at the end of each of its turns), or that happen at the moment the character becomes frightened (fear also causes a creature who fails their save to drop whatever they are holding). But if failing a save would cause the frightened condition, whether or not there are other effects, then a halfling gets advantage on the saving throw.
So: you roll a single saving throw, without advantage, and either take full damage and are forced to flee, or take half damage and aren’t. You do not save separately against the damage and the other effect, for this spell or any other, whether you have advantage or not, unless the spell specifically says so.
If your DM wishes to split saving throws so there are always separate ones for damage and additional effects, whether for the purposes of advantage applying or in general, that’s a big change to the standard rules. It will, among other things, slow down play and greatly affect the efficacy of spells. I’d recommend talking to them about the reasons for this change.