I think from what I’ve read that this is the case, but I’ve read some other info that’s made it less clear to me.
On the second part of the question regarding container engines, I’m pretty sure that may also be correct, and it kinda makes me wonder a little about risks of engine lock-in, but that may be a little out of scope.
Linux distro is entirely irrelevant. Almost always, the distro is just a particular way of packaging things.
But also, assuming you’re talking about “docker” containers (or rather, OCI containers), then no the OS running the container doesn’t matter. That’s the whole point of containers.
Also what do you mean exactly by “container engine”? Do you mean docker? Because “docker” containers can run on podman, kubernetes, etc