1) Can we buy a license of "VMware VSPHERE 5 ESSENTIALS KIT FOR 3 HOSTS" and use the license downgraded to unlock backups?
Yes you can, in the license section of MyVMware you can downgrade two of the available CPU sockets to vSphere 4.x to create a vSphere 4 license key.
2) Is "VMware VSPHERE 5 ESSENTIALS KIT FOR 3 HOSTS" what I need to use veeam backup to do incremental backups?
There is no requirement to purchase Essentials for this, you could just purchase a vSphere Standard license to the CPU count needed. so unless you are planning on adding hosts to your environment you are just burning cash. Veeams only requirement is that it has to be a costed version of ESXi due to a limitation of the Free versions API.
3)This allows me to use the license on three hosts simulatenously?
Yes vSphere Essentials Kit for 3 hosts will allow you to user upto three hosts with 2 CPU each.
4) If we buy a new server can I use a downgraded license (to vsphere 4) for the old one and one of the other for the new one supporting vsphere 5)?
Yes you can, vCenter 5 can manage vSphere 4 hosts
5) With this kit can I use vsphere essentials on three hosts at the same time?
Yes you can.