Automorphisms in the Dihedral groups

Let g be a group and $a \in G$. Define $\phi_a:G\rightarrow G$ by $\phi_a(g)=aga^{-1}.$ Now Let $G=D_4$ and $a=r$, where $r$is the rotation. We must show that $\phi_r: D_4\rightarrow D_4$. So show what it is mapped to.

So far I have, $\phi_r(r)=r$, $\phi_r(r^2)=r^2$, $\phi_r(r^3)=r^3$, $\phi_r(e)=e.$

Now for the reflections, I have defined $h$ as the horz reflection and $v$ as the vert reflection. I also have said $d_1$ is the diagonal reflection around the line y=-x if it were to be graphed. (I dont know of a way to say it better) and $d_2$ as the reflection y=x. So $\phi_r(h)=d_2$, $\phi_r(v)=d_1$, $\phi_r(d_1)=rd_1r=h$ and $\phi_r(d_2)=v$. Is this correct?

Lastly explain how there can be no automorphisms of $D_4$ so that $\phi(r)=f$, where $f$ is a reflection.

For your second question, an automorphism of $D_4$ is an isomorphism $\phi$ from $D_4$ to itself. So if $\phi(a) = b$, the orders of $a$ and $b$ must be the same in $D_4$. But $r$ has order $4$ and $f$ has order $2$.