Homogenous of degree one in utility function.
The way you show that $v(p,m)$ is homogeneous of degree one in $m$ is correct, but the reason why this implies that, $e(p,u)$ is homogeneous of degree one in $u$, is not very precise in your argument. For example, duality tells us
$$v(p,e(p,u))=u,$$
where $u$ is just a target utility level, but should not be $u(x)$ as in your proof.

Here is one possible way to proceed: Since $v(p,m)$ is homogeneous of degree one in $m$, it can be written as
$$v(p,m)=mv(p,1)=m\tilde v(p).$$
Applying the equality $v(p,e(p,u))=u$ gives
$$e(p,u)=\frac{u}{\tilde v(p)},$$
which clearly implies that $e(p,u)$ is homogeneous of degree one in $u$. You can use a similar argument to prove homogeneity of the Hicksian demand.

With all that said, I would suggest you prove the original statement directly using the definitions of expenditure function and Hicksian demand. For instance,
\begin{align*}
e(p,\lambda u)&amp;= \min p\cdot x ~~\text{ s.t. } u(x)\geq \lambda u\\
&amp;=\lambda\min p\cdot\frac{1}{\lambda}x ~~\text{ s.t. }\frac{1}{\lambda}u(x)\geq u\\
&amp;=\cdots
\end{align*}