I am looking for an example of a post-critically finite polynomial $P$ (i.e. all critical points have finite orbit), which has both the following:
a critical point on its Julia set (such as the polynomial $z^2+i$)
a super-attracting cycle, other than $\infty$, i.e. a periodic orbit $z_0\mapsto z_1\mapsto \cdots\mapsto z_{n-1} \mapsto z_0$ such that at least one $z_i$ is a critical point (such as the polynomial $z^2-1$ that has a $2$-cycle $0\mapsto -1\mapsto 0$)
Is there some known example from the multibrot families $z^q+c$ or any other example?
How about a post-critically finite polynomial with two critical points on the Julia set?
The polynomial $2z^3-3z^2+\dfrac{1}{2}$ has the desired properties. The critical points are at 0 and 1, with $$ 0\mapsto \frac12 \mapsto 0 \qquad\text{and}\qquad 1\mapsto -\frac12 \mapsto -\frac12. $$ The point $1$ lies on the Julia set itself. The following picture shows the filled Julia set for this function, with the two critical orbits marked in yellow and red, respectively.
To find this example, I first decided to look for a cubic polynomial with one critical point of period two and one critical point that maps to a fixed point. I also decided (without loss of generality) that the critical points should be at $0$ and $1$, so the polynomial should have the form $f(z) = 2az^3-3az^2 + b$ for some constants $a$ and $b$. I then looked for values of $a$ and $b$ that satisfy the two equations $f(f(0))=0$ and $f(f(1)) = f(1)$.
As for having two critical points on the Julia set, the third Chebyshev polynomial $4z^3-3z$ has the line segment $[-1,1]$ on the real axis as its Julia set, and its two critical points are $\pm\dfrac12$.