Yes.
There are no limits* to how high a pixie may fly, provided that they end their turn only one square above the ground. However, if they do end their turn above their altitude limit they fall with the consequences laid out for falling (1d10 damage/10 foot of drop).
*The theoretical maximum height a pixie may reach on their turn (with no extra movement trickery) is 8*3 + 1 or 25 squares high (run 8, trade standard for another run 8, action point for another run 8, +1 for starting height), that's 125 ft at the end of which they would fall. Falling says to subtract the creature's fly speed so that leaves us at 95 feet of fall, meaning 9d10 falling damage.
In D&D 5e, the daily rate of mounted overland travel is generally the same as on foot, because horses get tired and adventurers carry a lot of heavy equipment.
See the section Special Travel Pace in the DMG (p. 242–243). This section starts:
The rules on travel pace in the Player’s Handbook assume that a group of travelers adopts a pace that, over time, is unaffected by the individual members’ walking speeds. The difference between walking speeds can be significant during combat, but during an overland journey, the difference 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.
In the same section, the rule is:
- In 1 hour, you can move a number of miles equal to your speed divided by 10.
and then:
- For a fast pace, increase the rate of travel by one-third.
- For a slow pace, multiply the rate by two-thirds.
So an unencumbered horse with a speed of 60 could theoretically travel 6 miles in an hour at a normal pace. At a fast pace (a gallop), 8 miles per hour. That's "twice the usual distance for a fast pace", where "usual" means a creature with a speed of 30. This suggests that a riding horse with no rider, traveling alone, can cover 48 miles per day at a normal pace.
So the rule that "a mounted character can ride at a gallop for about an hour, covering twice the usual distance for a fast pace" seems to exist to allow for mounted travelers covering short distances quickly by using the mount's speed instead of "the usual pace", for up to an hour each day.
So, according to the rules, a traveler on a horse at a normal pace (3 miles per hour) will cover about 24 miles in an 8-hour day. If you make the horse gallop for an hour each day (fast pace for a horse being 8 miles per hour), that range increases to 29 miles. That's within the realm of what you would expect in real life, with a fast horse on good roads in fair weather.
Variant: Encumbrance
If you're using the encumbrance rule, a Riding Horse needs to be carrying less than 80 lbs of rider and equipment to get its full speed of 60. Loaded with between 80 and 160 lbs it has a speed of 50, and carrying between 160 and 480 lbs (its maximum carrying capacity) it has a speed of 30. A 200 lb adventurer in chainmail with a dungeoneering pack, longsword, and shield weighs in at about 325 lbs, so under this rule a horse's travel pace is usually the same as an unencumbered adventurer on foot.
Best Answer
Yes it is, but it does not tire
The phantom steed can not fly, as it
A riding horse has no fly speed, and so the phantom steed has no fly speed. Since it does not have a fly speed, and the spell does not state the steed is unaffected by difficult terrain, it is affected by difficult terrain on the ground.
Why then does it have an increased overland travel speed?
Several examples in the DMG with increased overland travel speeds cannot fly. We can observe that a steam-powered contraption does not fly, nor does a sailing boat. But they still do not travel at the normal rate, because they do not tire. The same applies to the phantom steed. The quoted section mixes two different reasons for increased overland travel speed:
In the case of the steam contraption the engine does not tire, in the case of the sailing boat, the wind does not tire, and in the case of the phantom steed, the magic does not tire. This is stated explicitly on page 243, further down in the same section:
There is one issue with the rule: if either not tiring or being able to fly can result in increased overland travel pace, should the combination of both (as provided by a flying carpet) not have a pace that is even larger? The rule does not say so. Maybe that was a design decision to keep the rule from getting too complicated. However, you could use the carpet to fly for 24 hours instead of the normal eight, which could result in a tripled overland travel pace (thanks to Ryan C. Thompson for pointing this out).
PS. Issues with the special travel pace rules
For what it's worth, the travel rules in the DMG are a bit wonky to begin with. While they maintain that this is travel at a special, faster rate, the formula to calculate that rate (pp. 242, 243 DMG) is:
A normal medium-sized character who can tire and cannot fly has a speed of 30. According to the Travel Pace table (PHB p. 182) that character's normal travel pace per hour is 3 miles and their daily travel pace is 24 miles -- exactly the same as calculated using these rules for special pace.
So that calculation does not result in travel pace that is faster relative to movement speed than what anyone could get, while it is presented as if it would be due to special advantages. The rule does lead to faster absolute travel pace if you have higher movement speed, as by the normal rules, travel pace would be independent of movement speed. There is, however, no mechanical speed-up for the ratio of normal speed to travel pace at all.
The PHB itself also states that normal roads are not difficult terrain and would not slow you down:
While that is at least consistent with the achieved travel speed, it means that the reason flying creatures are faster cannot be difficult terrain in the technical sense. It must be that there is no direct route, that the overland route is winding to avoid terrain features, while the distance as the bird flies straight is much shorter. Such may not be the case when you use a well-constructed road such as the one connecting Neverwinter and Waterdeep along the Sword Coast.
What is more, these speeds are too low compared to real world benchmarks. Using the rules, an eagle (fly speed 60) could travel 6 mph or 48 miles in a normal day of travel. However, in the real world an eagle can fly on average 100 miles in a day, or up to 250 miles when migrating. So the speed should be at least twice of what is given by the rule.
If we infer that the travel pace should be doubled because there is no difficult terrain to overcome for fliers (that normally would halve the speed), or because there is no exhaustion effect from repeated dashing for creatures that cannot tire (as suggested in the Chase rules on page 252 DMG), then to be internally consistent, the formula for creatures not tiring or being able to fly should be movement speed divided by 5 in miles per hour -- double the pace given. And possibly the rate for a flyer that also cannot tire should be movement speed divided by 2.5 in miles per hour - four times the pace given. All of that would be house ruling. (We use double pace in our home game, and had no issues with it.)