Is there any way I can use elevation contour lines to interpolate the approximate elevation of structures that are at ground elevation, e.g. manhole rims? If yes, is there a way to do the analysis for all my structures at once?
You can convert the contours to a DEM (raster) using Spatial Analyst Tools>Interpolation>Topo to Raster. Then extract the elevation of the DEM to the points that represent the structures using Spatial Analyst Tools>Extraction>Extract Values to Points. The finer the DEM (the smaller the grid size), the more accurate your structure elevation will be.