In a sense, yes. There are binary star systems. These consist of two stars orbiting each other, or more properly, orbiting their barycenter, their combined center of gravity. You might think of each of them as the other's satellite.
A moon
No, stars like the Sun stand alone and do not revolve around anything, so they are not natural satellites.
No. An eclipse occurs during the presence of a natural satellite
A natural satellite is a small body of rock or sometimes gas that orbits a planet.
the definition of a star is hot ball of gas, so it is a star.