Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.

**Claim**: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is a modular function for $\Gamma_{0}(25) \cap \Gamma_{1}(5)$. (This is straightforward to prove using properties of the slash operator.)

Using this claim, it's not hard to see that $\eta(\tau/5)/\eta(5\tau)$ is a modular function of level $5$, using the standard result of Gordon, Hughes and Newman about when an eta quotient is modular of level $N$. (This result can be found in the Wikipedia article about the Dedekind $\eta$-function.)

It remains to show that $\frac{\eta(\frac{\tau+1}{5})\eta(\frac{\tau-1}{5})}{\eta(\frac{\tau+2}{5})\eta(\frac{\tau-2}{5})}$ is a modular function of level $5$.

The Siegel functions $g_{(a_{1}/N,a_{2}/N)}$ can be used to build modular functions of level $N$. (See Section 5 of the article here by Amanda Folsom that gives a product expansion of the Siegel functions and criteria of Kubert and Lang that indicate when a product of Siegel functions is modular.) The product formula implies that
$$
g_{(0,a_{2}/N)}(\tau) = c \eta(\tau + a_{2}/N) \eta(\tau - a_{2}/N)
$$
for some constant $c$. The modularity criteria imply that
$$
h(\tau) = \frac{g_{(0,1/5)}(\tau)}{g_{(0,2/5)}(\tau)} = \frac{\eta(\tau + 1/5) \eta(\tau - 1/5)}{\eta(\tau + 2/5) \eta(\tau - 2/5)}
$$
is a modular function for $\Gamma(5)$. Now $h(\tau) = h(\tau + 1)$ and so $h(\tau)$ is a modular function for $\left\langle \Gamma(5), \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\right\rangle = \Gamma_{1}(5)$. Since $\Gamma_{0}(25) \cap \Gamma_{1}(5) \subset \Gamma_{1}(5)$, the claim above implies that $h(\tau/5)$ is a modular function for $\Gamma(5)$ and this proves that $\phi$ is a modular function of level $5$.

As a note related to your first question about $\phi^{24}$ being modular of level $5$ implying $\phi$ is modular of level $5$, Theorem 1 of Kubert and Lang's paper here is the following. Let $N$ be a prime power and $U_{N}$ be the group of modular units of level $N$ that are generated by the Siegel functions. If $g$ is a modular function and there is a positive integer $k$ so that $g^{k} \in U_{N}$, then $g \in U_{N}$.