Out of those 3 only PATH works on Windows.
Windows has a USERNAME environment variable, but depending on your needs you may need to combine it with the USERDOMAIN variable to get the complete username, %USERDOMAIN%\%USERNAME%.
Windows has a HOMEPATH environment variable but it contains only the directory for the home folder. Another variable, HOMEDRIVE, contains the drive the home directory exists on, so the complete home location is %HOMEDRIVE%\%HOMEPATH%. In powershell a read-only HOME variable is set pointing to %HOMEDRIVE%\%HOMEPATH%. The home directory is supposed to store user files. User specific applications, OS components, customization and configuration data should be stored in the USERPROFILE directory. This is very frequently but not always the same as %HOMEDRIVE%\%HOMEPATH%.