How can I figure out the managing HMC (and CSM server) of my LPAR?


To find the managing servers of your LPAR you can use the domain status information - the ctrmc subsystem must be active in order to do so. You can check with:

 # lssrc  -s ctrmc
 Subsystem         Group            PID          Status
  ctrmc            rsct             1151036      active
Is the subsystem indeed active you get the IP address of your HMC with:
 # /usr/sbin/rsct/bin/rmcdomainstatus -s ctrmc

 Management Domain Status: Management Control Points
   O A  0xccdfc3e608ad7624  0001  192.168.100.10
Is your LPAR also connected to a CSM server you would see two Control Points:
 # /usr/sbin/rsct/bin/rmcdomainstatus -s ctrmc

 Management Domain Status: Management Control Points
   I A  0x58016857defc1b87  0001  192.168.100.11  
   I A  0xccdfc3e608ad7624  0002  192.168.100.10
For newer versions of rsct you could also use lsrsrc to get this information:
 # lsrsrc -l "IBM.ManagementServer"
Resource Persistent Attributes for IBM.ManagementServer
resource 1:
        Name             = "192.168.100.10"
        Hostname         = "192.168.100.10"
        ManagerType      = "HMC"
        LocalHostname    = "192.168.100.23"
        ClusterTM        = "9078-160"
        ClusterSNum      = ""
        ActivePeerDomain = ""
        NodeNameList     = {"mylpar22-ext"}
resource 2:
        Name             = "192.168.100.11"
        Hostname         = "192.168.100.11"
        ManagerType      = "CSM"
        LocalHostname    = "mylpar22"
        ClusterTM        = "9078-160"
        ClusterSNum      = "10BF571"
        ActivePeerDomain = ""
        NodeNameList     = {"mylpar22-ext"}