1. You would need to purchase a license if you want Service and Support from VMware -You can with the Free Hypervisor - https://my.vmware.com/web/vmware/info/slug/datacenter_cloud_infrastructure/vmware_vsphere_hypervisor_esxi/5_1 - you would be able to add a purchased license at later to this server without having to rebuild. But please not there are limitations to the free license.
2.No you do not need Virtual Center - Virtual Center (vCenetr) allows you to manage multiple ESXi hosts and can provide additional service if your are licensed for it such as High Availability (HA) - the ability to recover from an ESXi host failure restarting the running VMs on the other hosts or Distributed Resource Scheduler - the ability to insure the necessary resources are being delivered to the VMs
3. I am not sure what the cost is I believe no more than $5000 -
4. vSphere is a marketing term that encompasses vCenter, ESXi and other VMware products - your will just need the lapproriate number of icenses
for ESXi which is licensed by physical core