Derive the cost function and supply function from production function

I didn't study economics, but am quite interested in the topic. I came to the question whether I could derive the supply curve / marginal cost function from the production function and I actually found a quite straight forward method, that I couldn't find online, so I would really appreciate if you could confirm (or correct) the result.

We are given the production function (we assume Cobb-Douglas), where capital is fixed and the only input is labor: $$Y(L)=L^a, t 1>0$$ The cost function is just wage times labor input: $$K=wL$$ We can manipulate our Production function, so that The labor supply is isolated and can then be substituted in the cost function: $$Y^{(1/a)}=L$$ Substitution gives: $$K=wY^{(1/a)}$$ where $$w=\frac{dY}{dL}=aA^{(a-1)}=a(Y^{(1/a)})^{(1/a)}=aY^{(1/(a^2))}$$ Plugged in into the cost function: $$K=aY^{(1/(a^2))}Y^{(1/a)}=aY^{(1/(a^2)+1/a)}$$ The supply function is equal to the marginal costs, so: $$t S=\frac{dK}{dY}$$ which is, dependent on a) a function of high degree. Eg. at $$a=0.3$$, $$S=0.3Y^{14,4}$$.

Is this result correct? I mean it is just easy manipulation so there should be nothing wrong about it. And we made no usage of any more “complicate” optimization as I found online.

And now that I've already registered here and took the time to pose a question: is there any “proof” (theoretically) or evidence that the compensation is always the marginal rate. I mean, as a manager I would always charge the average costs or pay my employees their average contribution, also in the “not equilibrium” state.

Please excuse if I made any language mistakes. I'm not a native speaker.