In general, it is backward compatible to run an older hardware compatibility version using a newer version of ESXi/Workstation/Fusion hypervisor. There are limits though as to how far back it can go but I don't think it would be a problem for ESXi 6.7 (natively supports version 14) to run version 8 or 10.
https://kb.vmware.com/kb/2007240
https://kb.vmware.com/kb/1003746
Yes, you have to power off the VM and upgrade them one by one. Hardware compatibility can also act as mask of CPU features. For example, hardware version 10 or earlier masks out Haswell CPU capabilities/features such as INVPCID instruction.
So if you have Haswell or newer CPU in the servers and don't have an EVC mask of Ivy Bridge or earlier, it would be better to upgrade the hardware compatibility even though you might not be looking at using specific hardware version features (e.g. secure boot, virtual NVMe in version 13).
The guest OS will also help you determine the urgency/priority to upgrade the hardware compatibility. For example, Windows 2008 cannot support the INVPCID instruction while Windows 10 can use the INVPCID instruction to mitigate the performance hit due to the Meltdown patch.