Suppose $p(x)$ is a polynomial with real coefficients that breaks through the $x$-axis at $14$, bounces off the $x$-axis at $-63$, and breaks through the $x$-axis at $57$. If $p(9+i)=p(-5+4i)=0.$

What is the smallest possible degree that $p(x)$ could have?

If $p(x)$ has no other roots than those described above, could the degree of $p(x)$ be 26? (yes or no)