When you say "biz kadınız" you're using kadın as a predicate nominative (fancy way of saying you're saying X = Y) -- we are women. In English you have to match the subject number to the object number, but in Turkish you don't.
"Biz kadınlarız" definitely is not a natural full sentence in Turkish. You could stretch it and say it's a sort of colloquial way to say "We women" as a phrase. Here, I'll pull a couple from the Googles. "Türkiye'nin En İyi Ekonomisti Biz Kadınlarız" -- Turkey's Best Economists are We Women" or "Yaşamı Yaratan Biz Kadınlarız" -- the ones who create life are we women. But it can't work to mean "we are women."
So is the general rule that if the plural-ness of a noun is implied by the ending from a different case you don't add on an additional ending? In this case, because we're in the nominative, you don't add "lar" because Iz indicates it's plural-ness (because we're talking about ourselves). Is that correct? Does this only apply to people (because it's a copula) or to all nouns? If I wanted to say "The dogs are happy" I would say "Köpekler mutlu" right? I saw note #4 said never use 'lar' in the third person plural for animals and things.
Thanks in advance.
Yes. The general rule of spoken Turkish is don't use "-lar" unless it's semantically fairly important to clarify it's multiples and nothing else in the sentence is doing so.
There's technically a rule in formal Turkish that animate third person plural subjects get marked with -lEr (Onlar erkekler vs onlar erkek BUT köpekler mutlu, unless you're personifying the dogs excessively :) ) but it gets dropped so frequently in normal speech that even Duolingo isn't consistent with that.
They are both gramatically correct, and I've seen the usage of them both. But Turks usually use "kadın-ız" (which is conjugated with the singular form of "woman") since the plural form is not needed for plural subjects in Turkish. Just like how it goes for plural stuff in general. Like we say "beş elma" (five apple) and not "beş elmalar" (five apples)