Is it possible to autostart a VMware virtual machine in background as a Windows service, and shut it down elegantly when Windows shuts down?

You can also use the vmrun command.

You can use a script to start a VM, for example:

vmrun -T ws start "F:\VMWare-VMs\S1.vmx"

and you add the script at Windows startup.

More info about vmrun: http://www.sysadmit.com/2016/11/vmware-workstation-vmru