Yes you can, as long as Veeam supports both the source and the target host ESXi version.

The other way around could be an issue if you have VMs with a newer hardware compatibility than the old host supports.

Yes it will be the hardware version of the VM that is most important but going from older 6.5 to 7.0 should not be an issue at all since the HW version will be lower anyway.

Yes you can do this.
Just be aware to check the version of VMware tools and the hardware version of the VM.
From 6.5 to 7 it should be no problem and you can change both after the restore. Vice versa it could be a problem...

Also just recommend doing this on a new version of Veeam B&R. Older versions may work, but it’s a good idea to push to the newer on the Veeam B&R Side. Meaning, the newest version of Veeam B&R that supports both versions of the vSphere platform.

A trick to keep in your pocket…  The Replication job is a great way to move from an old cluster to a new one, replicate and failover. Easiest way to leave behind bad design decisions of the past :)