Locus of roots of all convex combinations of two monic polynomials

Not true. Take $p=z^2-z$, $q=(z+0.1)^2-0.2^2$. Then the zeros of $p$ are at $0,1$, those of $q$ are at $-0.3,0.1$, so the smallest distance is between $0.1$ and $0$. However, $0.1$ moves towards $1$, not $0$ (as one would in fact expect from a picture).

Whatever the truth of the conjecture is (well, Christian's answer tells us...) the zeros of linear combinations of polynomials have been studied, see for example:

On the zeros of a linear combination of polynomials (Pacific Journal, 1966m Robert Vermes).