Assume I have linear equation system in $n$ variables and the solution is $x_1=x_2=\cdots = x_n$.
What properties must the matrix of the system have, or how do I prove that the system has such a solution?
I assume that you have an $n\times n$ linear system $Ax=0$, whose solutions are precisely $x_1=x_2=\cdots=x_n$. This means that the null space is the linear span of the vector $e=(1,1,\cdots,1)$. So the row space is the orthogonal complement of $e$. That means the sum of each row of $A$ must be 0, and that $n-1$ of these rows must be linearly independent.
No. You can solve these type of equations irrespective of A being a square matrix or even otherwise. In Ax = 0, consider that A is a general mxn matrix. If m>>n, then you have a redundant system of equations. You can solve these equations using either of normal equations(pseudoinverse), Singular value decomposition or QR decomposition. Look at it as a least squares problem. x is a value which minimizes the difference between b(here it is zero) and Ax.