What you might call a stateful NAT is really a 1-1 NAT, anything going out picks up an IP and anything retuned to that IP is routed back to the single address behind the NAT. Most home users a many to one source nat so their internal devices pick up a routable IP and multiple connections to a given dest are tracked by a source port map to route return traffic to the appropriate internal host.

Basically yes to what you said, but a port forward technically is a route map inbound to a mapped IP. You could have an ACL or firewall rule to control access to the NAT but in itself the forward isn't a true firewall allow.

Same basic result but if you trace a packet into a router without a port forward it'll be dropped before egress rather than being truly blocked. I think where some of the contention lies is that routing between private nets you have something like:

0.0.0.0/0 > 192.168.1.1 10.0.0.0/8 > 192.168.2.1

The more specific route would send everything for 10.x to the .2 route and it would be relayed as the routing tables dictate from that device. So a NAT in that case isn't a filter.

From a routable address to non-route 1918 address as most would have from outside in though you can't make that jump without a map (forward) into the local subnet.

So maybe more appropriate to say a NAT 'can' act as a firewall, but only by virtue of losing the route rather than blocking it.