What can be said about a matrix which is both symmetric and orthogonal?

For your first question, the answer is no. Every real Householder reflection matrix is a symmetric orthogonal matrix, but its entries can be quite arbitrary.

In general, if $A$ is symmetric, it is orthogonally diagonalisable and all its eigenvalues are real. If it is also orthogonal, its eigenvalues must be 1 or -1. It follows that every symmetric orthogonal matrix is of the form $QDQ^\top$, where $Q$ is a real orthogonal matrix and $D$ is a diagonal matrix whose diagonal entries are 1 or -1.

Can a matrix with the desired properties only contain the values -1,0 and 1 ?

For this part of your question every 3-D rotation matrix (it's orthogonal) about any axis ( defined by a unit vector $v$) by angle $\pi$ is symmetric.

You can generate plenty of them with Rodrigues' rotation formula which for a $\pi$ case takes simpler form $rot(v, \pi)= 2vv^T-I$ and they are not necessary consist only of $-1, 0, 1 $.