Sorry I meant high availability as in the ability to live transfer a VM to a different host without downtime or service interruptions.
Oh, my bad then. But yes, like Proxmox, LXD/Incus can do live migrations of VMs since 4.20 (2021 I believe). Live migration of containers can be done under specific circunstantes as well.
Are you using a container runtime in the LXC container? (i.e. docker or podman)
In some of them yes. At least under Debian as long as you've set security.nesting=true it will work fine.