Many of you would know of the Irwin-Hall distribution which states that if you have $n$ independent and identically distributed $U(0,1)$ variables, $U_1, U_2, U_3 \cdots$ and let $X = \sum_{i=1}^{n} U_i$, then the probability density of function of $X$ is: $${\displaystyle f_{X}(x;n)={\frac {1}{2(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x-k)^{n-1}\operatorname {sgn}(x-k)}.$$
Is there something similar that gives the cumulative density function of $X$? Also, is there a way to change the parameter for the variables to make it so that they are uniformly distributed from $(-1, 1)$?
Refer to Wikipedia for the CDF, which is
$$F_X(x) = \frac{1}{n!} \sum_{k=0}^{\lfloor x \rfloor} (-1)^k \binom{n}{k} (x-k)^n.$$
As for your second question, this is something you can derive yourself via a suitable transformation, namely if $X \sim \operatorname{IrwinHall}(0,n)$, we have $$Y = 2X-n,$$ thus $X = g^{-1}(Y) = \frac{Y+n}{2}$ and $$f_Y(y) = f_X(g^{-1}(y)) \left|\frac{dg^{-1}}{dy}\right| = \frac{1}{2} f_X(\tfrac{y+n}{2}),$$ where the density $f_X$ is the formula you provided.