From the server side, you could use the IIS web adaptor and enable logging in IIS.
From the client side, you could run Fiddler.
ArcGIS Server uses a variety of Apache software. If you look under C:\Program Files\ArcGIS\Server\geronimo for ".properties" files, you will find many. I believe that the ESRI position would be that these are internal components that should not be modified.
