1.) vCenter Server is basically a management application/service, which allows to centrally manage multiple hosts in an environment. The vSphere Client is used to connect to vCenter Server (or an ESXi host), and is simply a management interface.
2.) You can use the vSphere Client to connect to, and manage the ESXi host.
3.) vCenter Server has to be licensed, and - in order to manage ESXi hosts - the hosts need to be licensed too, i.e. you cannot manage ESXi hosts with the free Hypervisor license in vCenter Server. The lease expensive edition is an Essentials Kit, which contains a license for vCenter Server and up to 3 hosts. Anyway, if you only want a host with 3 VMs, and don't case about the limitations (up to 8 vCPUs per VM, restricted API access for e.g. backup applications) you can go with the free Hypervisor license.
André