The travel pace rules say they are meant to represent a group of travellers of differing capabilities. In effect, they are rules of averaging, in that they:

assume that a group of travelers adopts a pace that, over time, is unaffected by the individual members' walking speeds. . . . [D]uring an overland journey, the difference [i.e., in their speeds] vanishes as travelers pause to catch their breath, the faster ones wait for the slower ones, and one traveler's quickness is matched by another traveler's endurance.

DMG p. 242. The overall effect is to treat all such travelers as if they have a walking speed of 30 feet and calculate pace from there. For simplicity, let's focus on the hourly pace -- a "normal" (i.e., not fast or slow) pace given as 3 miles per hour. See PHB p. 181-182.

The rules also contemplate a mounted character traveling at double pace for an hour, because "many animals move much faster than humanoids." PHB p. 181.

The DMG also includes "special travel pace" rules to reflect a creature traveling via "magic, an engine, or a natural force (such as wind or a water current)," which wouldn't "tire the way a creature does" and might avoid "the types of obstructions found on land." See DMG p. 242-243. In that case, formulae for special paces are provided. The special formula for hourly pace is walking speed divided by 10.

These rules appear to leave a gap, however, regarding creatures traveling alone -- unhindered by slower companions -- but without the benefit of either a mount or a means of tireless locomotion. All else being equal, a rogue with Cunning Action can move significantly faster than other creatures by dashing. Nevertheless, that fast movement is still movement on foot; it is not "magic, an engine, or a natural force." And from a "what does the character sheet say" perspective, it doesn't change her walking speed.

If a rogue with Cunning Action (which can be used to dash) and a walking speed of 30 feet travels overland, what is her pace? Is it:

• 3 miles per hour, because the normal travel pace rules apply and
  Cunning Action does not affect them?


• 4 miles per hour, because Cunning Action represents a "fast" pace
  under the normal travel pace rules?


• 6 miles an hour, because the special travel pace rules apply and with
  Cunning Action she can move twice as fast as a character with a
  walking speed of 30 normally could?


• 6 miles an hour as above, but only for an hour because she is functioning like a galloping mount?


• 9 miles an hour, because the special travel pace rules apply, with
  Cunning Action she can move twice as fast as a character with a
  walking speed of 30 normally could, and she can make fullest use of
  her mobility because she is alone?


• 9 miles an hour as above, but only for an hour because she is functioning like a galloping mount?


• Something else?

Does the answer change if the rogue is affected by longstrider?

A rules-as-written answer gets bonus points, but any persuasively-reasoned answer is acceptable.

3 miles per hour

Actions in Combat are not relevant to the travel pace rules. Rogues are faster movers in combat thanks to Cunning Action, but so are most characters as a character's walking speed reflects:

Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round. This number assumes short bursts of energetic movement in the midst of a life- threatening situation. (PHB 181)

[courtesy of MisterB in the comments]

Cunning Action and the Dash action do not affect travel pace. Compare this to a feature like the Totem Warrior Barbarian's Elk Aspect of the Beast which directly mentions travel pace:

Whether mounted or on foot, your travel pace is doubled, as is the travel pace of up to ten companions while they’re within 60 feet of you and you’re not incapacitated (see “Adventuring,” for rules on travel pace). The elk spirit helps you roam far and fast.

What about Longtrider?

Longstrider would increase your travel pace most likely. Travel pace of 3 miles per hour is derived from the common walking speed of 30 feet. According to the Dungeon Master's Guide:

In 1 hour, you can move a number of miles equal to your speed divided by 10.

Since longstrider increases your speed, your travel pace also improves during the duration (which by default is 1 hour).

Why can't I just Dash?

Combats rarely last more than a couple minutes and require a character to do a lot of things that would be very physically taxing should they be done for extended periods of time. In the rules for Chases from the Dungeon Master's Guide this is made apparent by this section on dashing (mentioned in HellSaint's answer):

During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.

While this isn't directly related to travel pace it demonstrates that staying in combat mode for long periods of time (in order to Dash) could quickly result in exhaustion before you simply can't move further.

Constantly dashing is (probably) not viable

The DMG has a Chases section on p. 252, which contains a Dashing subsection. There, you can see that

During the chase, a participant can freely use the Dash
action a number of times equal to 3 + its Constitution
modifier. Each additional Dash action it takes during
the chase requires the creature to succeed on a DC 10
Constitution check at the end of its turn or gain one level
of exhaustion.

This is probably explained in the section relevant to chases because constantly dashing isn't usual during a normal combat, or during the usual day.

A DM could (and I personally would) rule that actually running (Dashing) just exhausts you in a long time travel and reduces how long you can actually travel in a specific adventuring day. See, for example, a marathon. They run at quite slow paces so they can actually finish it.

What I'm saying is: dashing during your travel is not something that will be happening, thus the Cunning Action is not relevant here.

Since this is irrelevant, we come back to 3 miles per hour.

RAW

You mention you would also like RAW answer. Well, RAW the explanation is easier: neither Dashing or Cunning Action state anything about increasing your travel pace, thus they don't.

Longstrider

Since it's a long duration spell (1 hour), it should affect the travel pace, since it directly increases your Movement Speed. As David Coffron mentioned, the travel pace is based on the movement speed

In 1 hour, you can move a number of miles equal to your speed divided by 10.

Longstrider adding 10 feet of MS would increase your travel pace by 1 mile per hour, resulting in 4 miles per hour. Note, though, that this would only apply for the first hour - and then either you would have to recast Longstrider or come back to 3 miles/hour.

Her Overland Pace Remains Unchanged

Dash does not affect the base speed, and only applies during encounters. Movement & Travel

Movement in Combat Doesn't Translate to Travel Speed

Combat is a hectic mess of adrenaline pushed actions to defeat an opponent and preserve one's own life. Movement speed in combat does not translate directly as one could not sprint all day long.

Short-Term Base Speed Increases Do Not Affect Travel Speed

Traveling for 8 hours in a day means that the increases afforded by short terms spells are going to be negligible in terms of the entire travel day.

Long-Term Base Speed Increases

A monk with unarmored movement or the recipient of longstrider 8 times a day would have an all day addition of 10 feet to the base speed. That gets the recipient 4 extra miles in the day for normal walking pace.