Question

A satellite of mass M moves in a circular orbit of radius R at a constant speed v around the Earth which has mass ME.

Answer 1

-- "Work" is the product of (force exerted on the object) x (distance the object moves in the direction of the force).

Since the orbit is a circle, the gravitational force toward the center is always perpendicular to the orbit.  The object never moves in the direction of the force.  If it did, it wouldn't be 'R' away from the center of the circle.

So the product of (gravitational force) x (distance in the direction of the force) is always zero.

-- Even if the orbit ISN't a circle . . . there are some parts of the orbit that aren't quite perpendicular to the gravitational force.  If the satellite is traveling through one of those parts AND getting closer to the central body, then gravity is doing positive work on the satellite.  If the satellite is traveling through one of those parts and getting FARTHER from the central body, then the satellite is the one doing positive work, and gravity is doing 'negative work'.  The work done by gravity ... and the work done by the satellite ... is zero over a complete revolution, although not zero at every point.

This is exactly the definition of a "Conservative Force" ... a force that does zero work through one trip around any CLOSED path.  Gravity is a conservative force, and so is the Electrostatic force.