Please see the PI Performance Monitor (PI PerfMon) interface. This interface can read all the performance counters such as memory usage and CPU counters provided by Windows and send them to the PI Server to be monitored and archived under corresponding tags. It should cover at least some of what you mentioned.
v2.3 and v2.4 AF Server only implement a single custom performance counter named "Health", which is documented in the PI AF 2010 R2 Installation and Maintenance User Guide on page 65.
The AF Server is a WCF service and WCF diagnostic performance counters can be enabled in AFService.exe.config. Search for "diagnostics performanceCounters" in AFService.exe.config.
Memory and CPU can be monitored via standard Windows performance counters. SQL Server performance counters are available. As a stateless web service, AF Server does not have a concept of "active client connection".
The AFSDK exposes a class that can report AF RPC metrics, which are displayed in PSE: PISystem::GetRpcMetrics( )
SQL Profiler can track communication between AF Server and SQL Server and it is possible to capture this communication and calculate min / max / average / sum of sql stored procedures execution times.