yes, HA will function properly if any one of the host goes down. since HA will restart the VMs on another host so the new CPU family features will be applied for the VMs when they were restarted. however, i would recommend you to enable EVC on the cluster to get the vMotion benefit.
Refer the below documents for EVC
Yes. As HA essentially resets the VM on another host, it will pick up the new CPU ID and boot up without any issues.
Can you look at getting some down time on the workloads to get EVC enabled?
You will need to edit the cluster settings and edit EVC mode then select any compatible mode from drop down list.
When you select any EVC mode, it will show you any incompatible in bottom pan.
and from your question, HA will still work but you may not be able to do live vmotion.
In HA, VM will restart onto another host.