When an excess charge is placed on an isolated conductor, it will distribute itself on the surface of that conductor so that all points of conductor whether on the surface or inside the conductor becomes to same potential. This is true even if there is a cavity inside the conductor.

If an isolated conductor is placed in an external electric field, all points of conductor still come to a single potential regardless of whether the conductor has an excess charge. The free electrons distribute themselves on the surface in such a way that the electric field they produce at interior point cancels the external field that would otherwise be there.

In electrical field, equipotential surfaces must

have decreasing potential in the direction of field

Uniform field means a constant field at every point and it is directed in a fixed direction for all points. Thus equipotential surfaces must be plane surface. The electric field is always perpendicular to equipotential surface and kts directed from higher to lower potential.