I don't have a complete solution, but the following may be helpful.

Change variables by $z_i = \sum_j y_j \xi^{ij}$ where $\xi$ is $m$-th primitive root of $1$.
Then the first line equations (I am using mod $m$ notation for the indices unless otherwise stated) become
$$
0=\sum_s z_s z_{2t-s} = \sum_{s,j,k} y_jy_k \xi^{sj+(2t-s)k}
=\sum_{jk}y_jy_k \xi^{2tk} (m\delta_j^k) = m \sum_k y_k^2 \xi^{2tk}
$$
$$=m\sum_{k=0...m/2-1} (y_k^2+y_{m/2+k}^2)\xi^{2tk}.
$$
This impies
$y_k^2+y_{m/2+k}^2=0$ for all $k$, so $y_k=\pm I y_{m/2+k}$, 
which are linear equations on $z$.

Similarly, we get linear equations on $z$ from the second and third line in the original post.
The problem is now to assure that these are of rank $m$ for any choices of signs above and any choices in the second and third lines. Good luck!