1. Yes, you can use any PC you like as long you have VMware CLI installed on it, Windows, Linux dosen't matter, vSphere CLI is available for both platforms
vSphere CLI for vSphere 6.5 - VMware {code}
2. Any distribution of Linux or Windows, matter of taste
3. The command is just the shutdown command, you need to create a scheduled task in Windows, or a cronjob in Linux that run the command on the time of your choosing
Windows: https://technet.microsoft.com/en-us/library/cc748993(v=ws.11).aspx
Linux: https://www.cyberciti.biz/faq/how-do-i-add-jobs-to-cron-under-linux-or-unix-oses/
4. Is starts with the host and the host sends a shutdown command to the VMs, if the host recives a heartbeat from VMware tools it will bring down the VM gracefully, otherwise it will just power them down.
5. Well, that depends on your router, or the client that sends the WOL package if it have support to shedule it.
WOL is pretty crappy and work when it wants to, but is the only thing available for the consumer market.
If you want a more reliable solution i suggest you take a look on a cheap server, Supermicro for example with an BMC/Lights-Out controller, those have the ability to power on on schedule by themselves.