Yes.
Yes, and they do. Not all of them have confirmed moons but some do. Pluto, for example, has four known moons.
The Sun doesn't have "moons". The term "moon" applies to satellites of planets or planetoids. An object orbiting the Sun is classified as a planet, dwarf planet, asteroid, comet, or a Kuiper Belt object.
Moons orbit planets. Planets (and dwarf planets) orbit stars.
As of now, the only planet known to have its own moon is Earth. Other planets in our solar system have moons but no planets of their own. In our solar system, moons primarily orbit around planets rather than planets orbiting around other planets.
Mercury and Venus. These two planets have no known moons.
There are a total of 214 moons orbiting planets in our solar system.
A moon? I don't think a planet orbiting another planet would be called a planet.
In our own solar system, the only planets with many moons are outer planets. It seems likely that if there were a large planet close to a companion star, the star's gravity would strip away any orbiting moons. However, we have no way of detecting the moons of any extrasoloar planets.
In our solar system, the dwarf planets Pluto and Charon orbit each other around a common center of gravity located in the empty space between them. As the only binary planets in our solar system, that makes Pluto and Charon share the title as having the most planets orbiting another planet. Planet like objects that orbit planets, without them orbiting each other, are called moons. The title for the planet with most moons, is passed back and forth between Jupiter and Saturn, as new (to us) and ever smaller moons are discovered. Currently, Jupiter holds the title with 63 moons.
There are no moons orbiting the planet Mercury.
That is the correct spelling of "planet" -- a large object orbiting a star. In our solar system, planetary bodies are generally classified by size and orbits as planets, dwarf planets, moons, or asteroids.
Tiny planets that orbit larger planets are known as moons or natural satellites. These moons are held in orbit by the gravitational pull of the larger planet. Some examples include Earth's Moon orbiting around Earth and Phobos and Deimos orbiting around Mars.