In general, you should probably turn on your router’s NAT even for IPv6.

No, you should not! NAT is not needed with IPv6 and you should never use it unless you really know what you are doing.

NAT is not a security feature, firewalls are, the default firewall rules from consumer routers are generally enough (allow outgoing, deny incoming except if it's an existing connection). And if you're concerned about others tracking hosts inside your network, the default settings of Privacy Extensions makes your device assign itself different IPs for outgoing connections every so often.