1) As I said before vSphere Client is management tool (GUI Interface) which is free to manage the ESXi host and vCenter server,
2) vSphere Client can be installed on any number of systems to manage the vSphere Infrastructure
The Only license what you need if for the vSphere suite..
I dont see any specific EULA for vSphere Client (http://www.vmware.com/download/eula/), You can contact VMware Sales team to get a confiramtion from them..