The previous answers responded to the main point how mergers between massive compact objects (i.e. black holes with black holes, neutron stars with neutron stars, etc) occur. We haven't yet observed a merger of a black hole with a neutron star but no doubt such mergers do occur. My answer provides some additional information about the history of detecting them, and also responds to the question about satellites.
As Florin notes, all mergers of massive compact bodies are rare events. However, the number of stars in the Universe is mind-bogglingly huge, so "rare event" x "huge population" means there are many such events happening across the Universe all the time, but we are only just beginning to detect them with our current level of technology. It's a very exciting time to be an astronomer!
The first Laser Interferometer Gravitational-Wave Observatory (LIGO) started operating in 2002, looking for the telltale gravitational wave signature from the merger of two black holes. After after 8 years it still hadn't detected a single g-wave. An advanced LIGO ("aLIGO") was then developed with about four times the sensitivity of the previous LIGO, and started collecting data in September 2015. Almost immediately (in fact, while they were still running tests on it prior to it formally being launched!) aLIGO detected the gravitational wave signature of the merger of two stellar-mass black holes over a billion light years from Earth. Three months later, aLIGO detected a second merger, and since then a further three black hole mergers have been detected. And in August 2017 aLIGO detected the collision of two neutron stars.
Further retrofitted enhancements to the LIGO network (the "A+" proposals) over the next decade are expected to nearly double aLIGO's sensitivity, so we can expect significantly more compact object mergers/collisions to be detected in future.
Regarding artificial satellites "colliding" with the Earth, we see this happen all the time - sometimes spectacularly! According to NASA, "During the past 50 years an average of one cataloged, or tracked, piece of debris fell back to Earth each day".
About half of Earth's artificial satellites are in orbit either within the exosphere - which extends from roughly 700 km to 10,000 km above sea level - or tens of thousands of km above the exosphere in geosynchronous orbit (such as some communications and GPS satellites). The exosphere is so extremely thin that the molecules - mostly hydrogen - rarely interact, so it no longer behaves as a gas, there is no "weather", and drag is minimal (although not zero). The orbits of satellites in the exosphere and above are mostly affected by solar wind, radiation pressure, variations in the Earth's gravitational field (e.g. from tall mountain ranges), and the gravitational influence of the Sun and the Moon, all of which can lead to an orbit slowly decaying (losing height).
Since 2002, the USA has required geostationary satellites to be moved to a graveyard orbit at the end of their operational life, to clear them away from operational satellites. This requires a small rocket boost to achieve an additional 300km in altitude.
However, about 500 operational satellites are instead in low earth orbit ("LEO") with an altitude of 2,000 km or less. LEO is the simplest and cheapest option for satellite placement, but the downside is that atmospheric drag becomes increasingly relevant once you're in the thermosphere from 80km to about 1000 km above sea level. The International Space Station orbits in this layer, at between 350 and 420 km; the Iridium communications satellites orbit within the upper thermosphere at 780km.
For LEO satellites, boosting to a graveyard orbit may be impractical. Satellites in lower orbits will lose altitude relatively quickly, and below 160km in altitude the orbit decays rapidly, eventually plummeting towards Earth. Small satellites will burn up on re-entry, but larger ones may not burn up completely and can hit the ground intact. Ideally, these larger satellites undergo a planned de-orbit (a controlled re-entry) to crash into a remote part of the ocean.
Wikipedia lists the largest space debris that has re-entered Earth's atmosphere. For example, the Salyut and Mir space stations were successfully de-orbited into the Pacific Ocean. The Skylab space station was slightly less successful, crashing spectacularly into the Australian outback in 1979.
And earlier this year, on 2 April 2018, China's Tiangong-1 space station made news headlines with its uncontrolled re-entry as a result of China losing its telemetry link with the spacecraft. The segments that failed to burn up on re-entry crashed into a remote part of the South Pacific, but this was entirely by accident rather than design.