Chapter 17 of The Science of Interstellar by physicist Kip Thorne (who was a consultant on the movie, and co-wrote the original script treatment) discusses Miller's planet and its orbit around Gargantua (the supermassive black hole seen in the movie, said in the book to have a mass about 100 million times greater than the Sun), and says:
Einstein's laws dictate that, as seen from afar, for example, from
Mann's planet, Miller's planet travels around Gargantua's
billion-kilometer circumference orbit once each 1.7 hours. This is
roughly half the speed of light! Because of time's slowing, the
Ranger's crew measures an orbital period sixty thousand times smaller
than this: a tenth of a second. Ten trips around Gargantua per second.
That's really fast! Isn't it far faster than light? No, because of
the space whirl induced by Gargantua's fast spin. Relative to the
whirling space at the planet's location, and using time as measured
there, the planet is moving slower than light, and that's what counts.
That's the sense in which the speed limit is enforced.
The "space whirl" he mentions refers to an effect called frame dragging, which can be thought of as space being whirled around the rotating black hole, which he discussed earlier in chapter 5. So from the point of view of distant observers, the planet completes an orbit once every 1.7 hours, so if they are using a coordinate system where the circumference is a billion kilometers this would be 588.24 million kilometers per hour, or about 163,400 kilometers per second, or about 55% the speed of light.
The quote above also answers your question "Would the planet appear to be orbiting the black hole very slowly from the perspective of the mother-ship, even though it is orbiting very fast?" In a way it depends on what you mean by "very slowly", but the answer is that the period of an orbit observed on the planet is much faster than the period observed from afar, by the same time dilation factor of around 61,000 that relates aging on the planet to the aging of faraway observers (since an orbit every tenth of a second is 61,200 times more than an orbit every 1.7 hours).
As for how they manage to navigate from one orbit to another, Thorne discusses this in chapter 7. Basically his answer is that although their rockets alone wouldn't be sufficient, they use gravitational slingshots past other massive objects in orbit around Gargantua, including smaller black holes and neutron stars. Quoting Thorne again:
In my science interpretation of Interstellar, the Endurance,
parked at ten Gargantua radii while the crew visit Miller's planet,
moves at one-third the speed of light: c/3, where c represents the
speed of light. Miller's plaet moves at 55 percent the speed of light,
To reach Miller's planet from the parking orbit in my interpretation
(Figure 7.1), the Ranger must slow its forward motion from c/3 to far
less than that, so Gargantua's gravity can pull it downwards. And when
it reaches the vicinity of the planet, the Ranger must turn from
downward to forward. And, having picked up far too much speed while
falling, it must slow by about c/4 to reach the planet's 0.55c speed
and rendezvous with it.
Fortunately, Nature provides a way to achieve the huge speed changes,
c/3, required in Interstellar: gravitational slingshots around black
holes far smaller than Gargantua.
Stars and small black holes congregate around gigantic black holes
like Gargantua (more on this in the next section). In my science
interpretation of the movie, I imagine that Cooper and his team make a
survey of all the small black holes orbiting Gargantua. They identify
one that is well positioned to gravitationally deflect the Ranger from
its nearly circular orbit and send it plunging downward towards
Miller's planet (Figure 7.2). This gravity-assisted maneuver is called
a "gravitational slingshot," and has often been used by NASA in the
solar system—though with the gravity coming from planets rather than a
black hole (see the end of the chapter).
This slingshot maneuver is not seen or discussed in Interstellar,
but the next one is mentioned, by Cooper: "Look, I can swing around
that neutron star to decelerate," he says.
To change velocities by as much as c/3 or c/4, the Ranger must come
close enough to the small black hole and neutron star to feel their
intense gravity. At those close distances, if the deflector is a
neutron star or is a black hole with a radius less than 10,000
kilometers, the humans and rangers will be torn apart by tidal forces
(Chapter 4). For the Ranger and humans to survive, the deflector must
be a black hole at least 10,000 kilometers in size (about the size of
Now, black holes that size do occur in Nature. They are called
intermediate-mass black holes, or IMBHs, and despite their big size,
they are tiny compared to Gargantua: ten thousand times smaller.
So Christopher Nolan should have used an Earth-sized IMBH to slow down
the Ranger, not a neutron star. I discussed this with Chris early in
his rewrites of Jonah's screenplay. After our discussion, Chris chose
the neutron star. Why? Because he didn't want to confuse his mass
audience by having more than one black hole in the movie. One black
hole, one wormhole, and also a neutron star, along with
Interstellar's other rich science, all to be absorbed in a fast-paced two-hour film; that was all Chris thought he could get away
with. Recognizing that strong gravitational slingshots are needed to
navigate around Gargantua, Chris included one slingshot in Cooper's
dialog, at the price of using a scientifically implausible deflector:
the neutron star instead of a black hole.
Finally, this may be more than you need to know, but one tricky aspect of this stuff is that in relativity there is no absolute notion of "speed", and likewise no absolute notion of the "circumference" of a black hole, these notions depend on the spacetime coordinate system you use to label physical events with position and time coordinates (and speed is then the rate of change in coordinate position with respect to coordinate time). In special relativity, which deals with the effects of high velocity in regions far from gravity, the notion that light always travels at the same speed, denoted by the constant c, is only true of a particular class of coordinate systems known as inertial frames; if you choose a "non-inertial" coordinate system such as Rindler coordinates in which a group of accelerating observers are treated as being at rest, then light's speed can be non-constant. In general relativity, which analyzes gravity in terms of the idea that mass curves the fabric of spacetime, all large-scale coordinate systems on curved spacetime are non-inertial, but if you zoom in a very small region of spacetime, in the limit as its size approaches zero you can define "local inertial frames" in that region, in which objects in free fall (not being acted on by any non-gravitational forces) are treated as moving at constant velocity, and such free fall observers using local inertial frames will measure the basic laws of physics in this small region to work the same way as in an inertial frame far from gravity (the equivalence principle), including the fact that light rays move at c. So in general relativity it's still true that light always has a velocity of c as measured locally by free-falling observers, and I believe that's what Thorne means in the first quote by "Relative to the whirling space at the planet's location, and using time as measured there, the planet is moving slower than light", despite the fact that the planet is completing an orbit ten times every second from the perspective of observers standing on it.