First, let's address the mass independence:

If we ignore the effects of drag from the air (which don't apply in space, and don't really apply to reasonably dense objects near the ground) all object fall to the ground at the same rate. A marble and a bowling ball, when dropped from the same height should reach the ground at nearly identical times. This is because the Earth's gravity exerts more force on more massive objects, and it take more force to make a more massive object move (and these two trends cancel out exactly!). The same is true for orbital velocities (orbiting is a special case of falling.)

Now let's address the speeds of different orbits:

The force that the Earth's gravitational field has on an object decreases as that object gets farther and farther away. It turns out that the force decreases with distance squared (so an object three times as far from the center of the Earth feels only 1/9 the force). So a bowling ball will take slightly longer to fall 10 meters when dropped from a very high weather balloon that it would from 10 meters above the ground (and significantly longer if dropped from as far away as the moon is).

If we assume that an orbit follows a circular path (a reasonable assumption for this discussion), then the length of the orbit is the circumference of the circle, and the radius of the circle is the distance from the object to the center of the Earth. If the distance of the object is increased by a factor of three, the path of the orbit also increases by a factor of three (because the circumference of a circle is equal to the radius times 2π).

So, if moving an object three times farther from the Earth decreases the force by a factor of nine, and increases the path it needs to take by a factor of three, it will obviously take much longer for the object to make one full orbit. Essentially, the farther away from the Earth the orbit is, the longer it will take.

An example: The International space station is only about 250 miles above the ground (~4250 miles from the center of the Earth), and orbits every 90 minutes. A Geostationary orbit is about 22000 miles above the ground (~26000 miles from the center of the Earth), and orbits every 24 hours. The moon orbits at about 240000 miles, and completes an orbit every 27 days.

Therefore for an object in geostationary orbit, if it begins to move away from the Earth, it will take longer than 24 hours to orbit, and if it moves closer to the Earth, it will take less than 24 hours to orbit, no matter how massive that object is (as long as the Earth is much, much more massive)

Hope this helps!