(1)

\begin{align*} \sum_{k=0}^{n-1}C\left(2n,k\right) & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n,k\right)+\sum_{k=0}^{n-1}C\left(2n,2n-k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n,k\right)+\sum_{k=0}^{n-1}C\left(2n,2n-\left(n-1-k\right)\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n,k\right)+\sum_{k=0}^{n-1}C\left(2n,n+1+k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n,k\right)+\sum_{k=n+1}^{2n}C\left(2n,k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{2n}C\left(2n,k\right)-C\left(2n,n\right)\right)\\ & =2^{2n-1}-\frac{1}{2}C\left(2n,n\right) \end{align*}

(2)

\begin{align*} \sum_{k=0}^{n}C\left(2n,k\right) & =\sum_{k=0}^{n-1}C\left(2n,k\right)+C\left(2n,n\right)\\ & =2^{2n-1}-\frac{1}{2}C\left(2n,n\right)+C\left(2n,n\right)\\ & =2^{2n-1}+C\left(2n,n\right) \end{align*}

(3)

\begin{align*} \sum_{k=0}^{n-1}C\left(2n-1,k\right) & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n-1,k\right)+\sum_{k=0}^{n-1}C\left(2n-1,2n-1-k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n-1,k\right)+\sum_{k=0}^{n-1}C\left(2n-1,2n-1-\left(n-1-k\right)\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n-1,k\right)+\sum_{k=0}^{n-1}C\left(2n-1,n+k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{n-1}C\left(2n-1,k\right)+\sum_{k=n}^{2n-1}C\left(2n-1,k\right)\right)\\ & =\frac{1}{2}\left(\sum_{k=0}^{2n-1}C\left(2n-1,k\right)\right)\\ & =2^{2n-2} \end{align*}