I assume you're referring to planetary systems? In principle, sure, you can have smaller planets orbiting a single larger planet-like object. In fact, we have systems like these in our own solar system: Jupiter and Saturn have tons of moons, so each of them look like a mini planetary system! Even the Earth has the moon orbiting around it, so we look like a planetary system with only one planet! Of course, it's true that Jupiter and Saturn themselves are orbiting the Sun, so what you're probably asking is if it's possible to have a planetary system like Jupiter and its moons which is not orbiting a star. As far as I know, I don't think these sorts of planetary systems can be formed the way solar systems are usually formed. That's because solar systems usually form when a large cloud of hydrogen and dust collapses under its own gravity: when enough of the hydrogen has collapsed, it starts to undergo nuclear fusion and turns into a star, and the rest of the hydrogen and dust in the cloud ends up forming planets orbiting the star.

However, it is possible for solar systems formed in this way to sometimes eject or "kick out" one of their planets. If this happened to a planet like Jupiter, which has many moons orbiting around it, the end result would be a solitary planet with moons, which looks like one of the planetary systems you're referring to.

Unfortunately, it's hard for us to know if these sorts of systems exists, and if they do, how many there are, because without a star to give off light, they'd be almost impossible to see! So everything I've said is mostly my guesswork, and is not based on observational evidence.

A massive planet or other massive body can acquire other smaller planets, moons, and other masses into its orbit. It doesn't meet the definition of a solar system, which would have to have a star at the center, but gravity is gravity and it applies to planets as well as stars. In fact, it is indeed the great mass of our sun that causes the planets to into orbit with it. Our sun is much more massive than any of the planets in our solar system.

This is an interesting question. While it has been discovered that there are planets that are larger than our sun outside of our solar system, I don't believe that any planetary systems with a planet at the center have been found.

It appears that planets need the debris surrounding either a protostar (a precursor to a star in nebula) or a dead star (after an explosion) in order to form.

Regarding the second part of your question: such a planet would have to be extremely dense or extremely massive. This phenomena of a body collapsing on itself is something that is seen in supergiant stars that undergo supernovas, or white dwarf stars that have exceeded what is called the "Chandrasekhar limit," the limit above which the electron degeneracy pressure in the dwarf star's core is not enough to balance the star's own gravity. The white dwarf stars undergo collapse, and turn into another object, such as a neutron star or black hole.

You can have planetary systems in orbit around even black holes, so the answer is yes. We've seen planets in orbits around many different sizes of stars, now, including pulsars, indicating that the planets somehow survived (or were created from the debris from) a supernova explosion. The closer a planet is to its parent star, the faster it orbits, but if it has enough angular momentum, it can still orbit. Large stars tend to have large radii, though, which limits how close a planet could be; Earth and even Mars, for example, are inside the radius of the star Betelgeuse, but Jupiter isn't, and there could easily be a planet in Jupiter's orbit around Betelgeuse. The most voluminous red giants have radii that extend past the orbit of Saturn, but not to that of Uranus or Neptune, so even they can have planets.

The planets orbit the sun because the sun has mass, not because the sun is a star. Anything that has mass can have satellites orbiting it. We see this even in our own planetary system; after all, the moon is in orbit around the Earth just as the Earth orbits the sun. If a planet with moons like the Earth were ejected from the solar system, those moons would go with it.

Lastly, black holes have gravity because they have mass, just like anything else. Because black holes take up so little volume, it is possible to get much closer to a black hole than it is any other kind of star, and as a result it is possible to get much more powerful gravitational fields from a black hole than from anything else in the universe, but only because the strength of gravity increases the closer you are to the gravitating object. If you were to remove the sun from our solar system and replace it with a black hole of the same mass, the Earth would continue to orbit the black hole in the same 365 days that it takes the Earth to orbit the sun.