Do Kepler's laws of planetary motion apply also to satellites in orbit around the Earth?
I would assume yes, but I'm not sure why...?
The laws of planetary motion and orbits are underpinned by Newtonian Physics and Kepler's Laws. These physical laws apply to everything in the universe and, as such, apply equally to the motion of planets and the motion of artificial satellites.
The major difference when calculating orbits for an artificial satellite is that it's very common to ignore its mass. This is because the mass of any satellite is many orders of magnitude smaller that the body it is orbiting and its effect is negligible.
Note - I am ignoring relativistic and quantum mechanical effects because I think they'd unnecessarily confuse an answer to this question
The laws of planetary motion apply not only to the planets of our Sun, the apply also to the Moon of our Earth, to moons of other planets and to satellites in orbit around Earth or Moon. Also to satellites of other planets.
But perfect elliptical or circular orbits require a so called two body problem, a star and a planet only. The influence of other planets should be neglectable and the diameters of the star and the planet should be very small compared to their distance. The mass of the planet should be much smaller than that of the star.
These laws should be valid also for stars with extrasolar planets.
Kepler's laws do a fairly good job of explaining satellite motion around the Earth to a limited level of accuracy for a short time, but you really have to add in more computation to predict their orbits for years, months, or even for weeks for the lowest ones.
Satellites in the lower parts of low Earth orbit, or LEO loose altitude rather quickly. The ISS has to regularly boost itself with engines to regain altitude. Space junk and other satellites will all tend to lower and then burn up over a period of years or decades and there's nothing in Kepler's laws that addresses this.
Another big and important effect is the shape of Earth and its gravity field. Since the Earth rotates, over time it has reached an oblate shape. The equatorial radius is about 6378 km, while the polar radius is only 6357 km. This oblateness means the gravity field is stronger near the equator than the poles, and while
Kepler Newton did do some work thinking about this problem, it's not covered by Kepler's Laws proper.
Sun synchronous orbits use this oblateness to precess their orbital planes around the Earth's axis once a year, so that they can match the direction of the Sun. Kepler's laws can't do that, but it's really handy and many Earth satellites use this constantly.
This precession was first experimentally confirmed by measurements of the doppler shift of radio signals and visual positions of the first few artificial satellites that orbited the Earth, Sputnik, Vanguard, etc. Read more about that in the links included in this answer.
Earth's gravity is also quite lumpy at a much finer scale and very careful measurements of the movement of satellites have helped engineers and scientists search for oil, understand distribution of material below the surface, and even measure the movements of the Ocean's tides.
Another problem is that the communications satellites in Geostationary orbits or GEO tend to move over time. Even if their orbital planes start exactly in the Earth's equatorial plane, their planes would start tilting right away if they didn't have tiny rockets or boosters to do what is called "station keeping" to stay over the right longitude but also to keep parallel to the equator.
According to the subsection titled orbital stability:
A combination of lunar gravity, solar gravity, and the flattening of the Earth at its poles causes a precession motion of the orbital plane of any geostationary object, with an orbital period of about 53 years and an initial inclination gradient of about 0.85° per year, achieving a maximal inclination of 15° after 26.5 years. To correct for this orbital perturbation, regular orbital stationkeeping manoeuvres are necessary, amounting to a delta-v of approximately 50 m/s per year.
A second effect to be taken into account is the longitudinal drift, caused by the asymmetry of the Earth – the equator is slightly elliptical. There are two stable (at 75.3°E and 252°E) and two unstable (at 165.3°E and 14.7°W) equilibrium points. Any geostationary object placed between the equilibrium points would (without any action) be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation. The correction of this effect requires station-keeping maneuvers with a maximal delta-v of about 2 m/s per year, depending on the desired longitude.