Looking at central difference coefficients, there is a clear pattern on the order of accuracy and the number of coefficients. For the first derivative with a 3pt stencil, the coefficients are given by $\{-1/2,0,1/2\}$ and the order of accuracy is $O(\delta)^2$.
If I were to free up the coefficients, while retaining the antisymmetry, such that the coefficients become $\{-a,0,a\}$, would the order of accuracy drop to $O(\delta)^0$, a constant? From what I gather, the error analysis at this level would be related to the mean value theorem? How would I formulate the error analysis in relation to the choice of the coefficient $a$?
Consider the Taylor series of your function $$ f(x)=c_0+c_1x+c_2x^2+O(\delta^3). $$ (The $O(\delta^3)$ can be made more rigorous using Taylor's theorem as described here). Then $f'(0)=c_1$, so $$ \frac{-af(-\delta)+af(\delta)}{\delta} =2ac_1+O(\delta^2) =f'(0)+(2a-1)c_1+O(\delta^2). $$ Taking $a=\frac12$, this is $f'(0)+O(\delta^2)$, but otherwise the difference is of constant order as you say.