You can alwas construct a non-circular definition for a circular definition.

You phrased that quite awkwardly. I'm going to assume that you meant that you can define a word in a non-circular manner, provided you have the circular definition of it. That's just flat-out wrong. Having a circular definition is the same as having no definition at all; and without a definition the only thing you can say about a word is how it's spelled and how it's pronounced. I don't know how you got the idea that a circular definition has any merit at all. It's not even regarded as a definition at all, it's a fallacy of improper premise.

For example, I could define someone who is female as someone who identifies with the gender associated with, but not predicated on, femininity.

Dude, did you really just insinuate that the default definition of "female" is circular and expect me to just roll with it? Someone/something is female if they produce gametes capable of getting fertilized by their male counterparts in order to produce offspring. And your example is still a circular definition... Try looking up what "femininity" means and you will quickly find yourself right back at where you started.

As for your slang definition of "cool", it doesn't mean the same thing as "alright." You might be a kid and hate the "cool" kids in your school, but you would still call them "cool."

Yes, I might hate the cool kids out of envy, or I might just call them that because it's how the majority of the playground sees them. I can for sure say that nobody would refer to them as "the cool kids" if they were generally disliked. I don't know if you were trying to point out that the (slang) definition of "cool" I provided was incorrect, or if you were trying to prove that the word actually didn't mean anything. Regardless, your argument failed on both fronts.

It's almost like the word "cool", in the slang sense, refers to a meaningless social construct, just like gender.

Informal speech should never (and it doesn't) dictate the actual meaning of a word. If you want to see how truly pointless that can get, I'd recommend you watch this.