It's recommended to install Power CLI inside vCenter server (6.0.0 - 3634793) or it's better to exclude it and install Power CLI into another server?
Yes you can install either on vCenter itself or any other Windows system. Make sure pre-requisites are met such as .Net, powershell etc.
What version can I install with vCenter 6.0.0 ?
You can install up to date version of PowerCli to work with vCenter 6.0.0 or either any of below ones
10.1.0
10.0.0
6.5.4
6.5.3
6.5.2
6.5.1
6.5.0
6.3.0 - vSphere PowerCLI
6.0.0 - vSphere PowerCLI
Ok, but what advantage to install latest version if I have 6.0.0 vCenter? Perhaps I'll get new functions/features which I will m not able to use them because my vCenter is not updated?
I thought to install 6.3 or 6.0.0 latest version, what do you think?
That's true. However the latest version of PowerCli would have all the cmdlets which are in version 6.3 so you complete control over whatever you will have in latest. I would prefer to install latest version when I wan't to explore new features and soon I am ready to upgrade vCenter version, else equivalent version is enough
Check out Which Version of PowerCLI is Right For Me? - VMware PowerCLI Blog - VMware Blogs