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I am creating 3D surface below the earth's surface using ArcGIS and then I need to find distances to the plane from the surface. Which coordinate system shall I use? 3D interfaces like IProximity3D and IArea3D look like they work only with Projected coordinate systems which are 2-dimensional. I am new to this area and need direction to decide and use.

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You'll need to define both a horizontal coordinate system (could be projected, like UTM) and a vertical coordinate system (like NAVD88, making sure to be in the same units as your horizontal).

A vertical coordinate system expresses elevations relative to a baseline, usually a mathematical representation of the Earth's surface. Your measured elevations would be negative, since you would be measuring below the surface.

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