Hello Alice -
I spoke with Michael Seybert in Esri customer service. He indicated that he works with your CIO (Teresa Luna?) ... Michael should be contacting you or someone in the Idaho OCIO soon.
I can confirm that the ArcGIS GeoEvent Extension for Server is in fact an extension for the ArcGIS Server product. It was first introduced as the "GeoEvent Processor" with the 10.2 product release of ArcGIS and renamed shortly thereafter to more appropriately identify it as an extension to the ArcGIS Server product.
GeoEvent is licensed separately from ArcGIS Server as an add-on / extension. It is Esri corporate policy to not discuss licensing terms or quote product costs through the forums or blogs. An organization's licensing is generally negotiated taking in to account the number of licenses and products you need to support your enterprise.
You might find the following links helpful:
If you do not hear back from Michael in a couple of days, please give him a call at 888-377-4575.
Best Regards --
RJ