Retreat is for when you all need to run!
Fighting withdrawal is always better for disengaging from combat, if it will actually do the job. A withdrawal only works if the opponent is unable or unwilling to follow to maintain the combat engagement – if they do, you've wasted a turn failing to disengage.
The other purpose of a withdrawal is to back into a more favourable position – into a doorway or narrow hallway, for example. However, if that choke point is beyond ½ your movement, then you have a dilemma – a fighting withdrawal won't get you there in one turn. If time is of the essence, do you try to do it in two turns of withdrawal, or do you break engagement to flee to the superior position and risk a strike at your back?
The difference between them is always that, sometimes, those few feet of difference can be worth the risk. If they're not, then a Retreat action isn't worth considering.
And yes, the monster gets its attack. Turn order is merely a game convenience for play simplicity, and for representing a slight combat advantage, but really every attack (by creatures that survive the whole turn) are considered part of a simultaneous mêlèe. The retreat grants that extra attack – as GM you can roll it at the time of retreat, or on its initiative turn, and the difference is a matter if convenience rather than whether the target is still viable. Think of it as happening "sometime" during the turn when the retreating character exposed themself to danger by recklessly turning tail, but do the actual roll and damage whenever is most convenient.
Tracking is mentioned only briefly, in/around the pages you mentioned, in both the PHB and the DMG. And as you pointed out, in the PHB and in the DMG it doesn't specify tracking as having any maximum speed.
On PHB page 181, under Travel Pace:
A fast pace makes characters less perceptive, while a slow pace makes it possible to sneak around and to search an area more carefully (see the "Activity While Traveling" section later in this chapter for more information).
This sentence, depending on how you read it, could mean that any "Activity While Traveling" (which is the section Tracking falls under) can only be taken while moving at a slow pace:
... a slow pace makes it possible to to search an area more carefully (see the "Activity While Traveling" section ...
However, several activities specify how they operate under different speeds, meaning this interpretation doesn't hold up.
Additionally, Foraging, another Activity While Traveling that points to full rules in the DMG, has the following statement (DMG page 111):
Characters can gather food and water as the party travels at a normal or slow pace.
It makes sense that Tracking would have a similar statement, but it does not, so for the time being there's no RAW restriction on tracking at a fast pace.
You can, however, interpret it as being RAI due to Foraging or due to the negative implication of the Barbarian ability.
Best Answer
Yes, characters can move more than 20 feet in a round and still hide
The travel pace you are referring to is for overland travel, according to PH 181:
The rules for combat rounds state that (PH 189):
If your speed is 35 feet (for example, a wood elf) you can move 35 feet and make a Dexterity (Stealth) check to hide as your action. Some characters can move even faster than their speed would indicate and still make a Dexterity (Stealth) check to hide.