Is it correct to say that falling object are standing still?

I would say Brian Cox is being too cryptic. He is stating what is known as the Principle of Equivalence. In pure general relativity, gravity is not a force. It is the curvature of spacetime causing objects to obey the geodesic equation. This is a geometrical feature: the geodesic equation has no mass dependence. In free fall, the objects are unaware of their acceleration. In their frame the objects are at rest with respect to the rest of the universe. I think he is just saying that objects at rest behave the same. It's definitely not how the EP is usually stated.

EDIT: The title of the question has changed. A falling object (assuming complete free fall, i.e. no air resistance) does not experience a gravitational field. Suppose you are in a box and are dropped from a great height above the Earth. You want to test if you are moving. (Rather, you want to test if you are accelerating. Special relativity tells us we cannot test for absolute motion. Assuming a gravitational field that is sufficiently constant in a sufficiently small region of spacetime, the question of absolute motion is meaningless.) So you fire a laser from one side of the box to the other. If you are accelerating, the laser will appear to "curve." If you are in free fall, however, the photons from the laser will hit exactly where the laser is pointed. This is exactly the behavior of a laser one would expect moving at constant velocity in flat spacetime (i.e. no gravity). We then Lorentz transform to a motionless frame (the rest frame of the box). Thus free fall is in a sense equivalent to being stationary in a gravity-free spacetime.

In general relativity, the gravitationally free-falling objects are inertial, while you standing on the Earth's surface are accelerated upwards by the force provided by the floor you're standing on. That is why you see the objects as accelerating downwards--because you are in an accelerated frame.

Therefore, your understanding of mechanics on this probably isn't wrong, yet Brian Cox's statement is still correct. You see inertial forces in Newtonian mechanics, e.g., in a rotating frame you have the centrifugal and Coriolis forces, which enable you to pretend that you're not in an accelerated frame and set up Newton's laws as usual, just with those extra terms. Conceptually, what GTR does to Newtonian $\mathbf{F} = m\mathbf{g}$ is simply say that gravitational force is an inertial force too, i.e. you can make it vanish by taking a local inertial frame, but it is alright if you pretend that you're inertial as long as you add it to the force equations (which Newtonian mechanics does anyway).

Quantitatively, of course, the Newtonian limit of GTR is quite a bit more involved, but the result is that Newtonian mechanics is a very good approximation in many circumstances. (In the GEM formalism for weak-field GTR, the centrifugal and Coriolis forces are themselves just special cases of gravitoelectric and gravitomagnetic fields.)

Good Question.

In classical mechanics, Newton's first law says that an object that experiences no force is at rest.

Of course this isn't quite correct when you look this up, as there is a qualification: or it moves at uniform velocity in a straight line.

But this is an artifact of Galilean relativity: imagine you're standing next to a sphere in space; and you're both at rest; now suppose you give the sphere a push; and it moves away at some uniform speed in a straight line; but because of Galilean Relativity, it's equally possible to imagine that the sphere is at rest and not moving, and you're the one that's moving.

What this means is that we have to expand the notion of rest so that it includes this phenomena.

In GR, it's the same except now the sphere is moving along a geodesic, a 'curved straight line'; this means it looks to us as though it is accelerating: which is exactly what we see.

What this doesn't explain though is why the earth being so close to the feather and cannon-ball shouldn't fall along with them so they all three move along in the same motion; and the feather and the cannon-ball wi never catch up with the earth.

This is because we're using the earths mass to curve the space around it; it's a local picture.

If we took the picture from the Sun, then yes, it does look like the earth, the cannon-ball and the feather are 'falling' together.