Is work required to move a negative charge towards a negatively charged object?

If they can, the electrons (we assume) will move over the surfaces so that they tend to the far side of each object. I suspect you suspected that. It's the repulsion of like charges that does it.

And, no, the electrons will not typically flow from one object, across the contact area, and into the other object. Again, the force is away from the contact point for both bodies. BTW: when the two objects are not touching and influencing the other, the electrons will distribute as best as they can over the surface of each one.

Why? Because the electrons keep being pushed from the inside out by the E fields and that "out" shoves them onto the surface. Note that wires carrying current might very well carry most of that current on the surface of the wire rather than internally.

For the same reason, the electrons are pushing outward, toward the surface as they flow to make up the current.

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