Best OS to run docker?
I just installed proxmox on a nuc and I would like to run jellyfin, nodered, home assistant, navidrome, logitech media server and a few other containers. What would be the best way to get these applications running?

If low overhead is desired you could give alpine Linux a try. It’s really easy and quick to install.
+1 for alpine linux, very low footprint.
I use Debian 11. Ubuntu's fine too.
Do you run these apps within the os or in docker?
Do u know how i can add shares from my syno to proxmox to the give out to my vms? It's only going to be read only smbs
LXC always
Personally some of those would benefit from being installed outside (without) of docker. I believe everyone forgets that docker ADD yet another layer of abstraction with adds over head.
I do acknowledge that docker makes the implementation of these and others simpler. But at a cost.
Debian or redhat family.
I suggest you to look at CT. Or even for jellyfin there is a LXC to make your life easier.
The best is probably alpine, but if you're already comfortable with a specific distro, it's all Linux, it'll work fine. I like apt so I use Ubuntu LTS.
You can either run docker nested in a LXC or in a VM. The latter is much easier, but uses more resources.
You could also run docker directly on bare-metal Proxmox debian, but that is a bad idea.
I use Ubuntu with a GUI. It is a bad idea in terms of performance, but it allows me to connect remotely for some stuff where I am not comfortable using the command line.
Good old Debian would be a much lighter and better base.