= Solution
Let $\mu_N=[PSL_2(\mathbb Z):\overline{\Gamma(N)}]$, where the bar denotes the effective projective image. This is the degree of the <modular curve> projection, rather than the full $SL_2$ index when $-I$ is not in $\Gamma(N)$.
Reduction modulo $N$ is onto $SL_2(\mathbb Z/N\mathbb Z)$. One proof uses the <Chinese remainder theorem> and elementary <matrices>: over $\mathbb Z/p^r\mathbb Z$, a unimodular first column has a unit entry, so elementary row operations reduce a determinant-one <matrix> to the identity. Each elementary <matrix> lifts integrally. Over $\mathbb F_p$, the first column has $p^2-1$ choices and the second has $p$ choices with prescribed <determinant>, giving $p(p^2-1)$. Each subsequent prime-power lift has $p^3$ choices, because the determinant-one linearized condition is one trace equation on four entries. Hence
$$
[SL_2(\mathbb Z):\Gamma(N)]=N^3\prod_{p\mid N}(1-p^{-2}).
$$
For $N=1,2$, the subgroup contains $-I$; for $N\geq3$ it does not. Consequently
$$
\mu_1=1,\qquad\mu_2=6,\qquad
\mu_N=\frac{N^3}{2}\prod_{p\mid N}(1-p^{-2})\quad(N\geq3).
$$
For $N\geq2$ there are no effective elliptic <stabilizer subgroups> in the subgroup. Indeed, write $\gamma=I+NB\in\Gamma(N)$. The <determinant> condition gives $\operatorname{tr}\gamma=2-N^2\det B$. A noncentral elliptic element in $SL_2(\mathbb Z)$ has trace $-1,0$ or $1$, none congruent to $2$ modulo $N^2$ for $N\geq2$. Thus the projective action is torsion-free.
All <cusp widths> are $N$. At infinity, $T^h$ lies in the effective subgroup exactly when $h$ is divisible by $N$: the alternative $T^h\equiv-I\pmod N$ can only occur for $N=1,2$, and for $N=2$ it imposes the same condition. Every rational <modular cusp> is a modular translate of infinity, and the subgroup is normal, so the same width holds there. The sum of <modular cusp> <analytic ramification indices> equals $\mu_N$, giving $\mu_N/N$ <modular cusps>.
The base $X(1)$ is a sphere. For instance $j=E_4^3/\Delta$ is a <meromorphic function> on it with a single simple pole at the <modular cusp> and no other poles, using the nonvanishing of $\Delta$ proved above. It therefore defines a degree-one map to the <Riemann sphere>, making the base <genus> zero.
The only branch values of $X(N)\to X(1)$ are the two elliptic orbits and the <modular cusp>. For $N\geq2$, the numbers of points above them and their <analytic ramification indices> are respectively $\mu_N/2$ with index two, $\mu_N/3$ with index three, and $\mu_N/N$ with index $N$. Applying the <Riemann-Hurwitz formula> directly,
$$
\begin{aligned}
2g(X(N))-2
&=-2\mu_N+\frac{\mu_N}{2}(2-1)+\frac{\mu_N}{3}(3-1)+\frac{\mu_N}{N}(N-1)\\
&=\mu_N\left(\frac16-\frac1N\right).
\end{aligned}
$$
This is the <elliptic and cusp ramification for principal level> calculation. Level one is the identity projection. The complete result is
$$
\boxed{g(X(1))=g(X(2))=0,\qquad
g(X(N))=1+\frac{N^2(N-6)}{24}\prod_{p\mid N}(1-p^{-2})\quad(N\geq3).}
$$
For example the genera for $N=3,4,5,6,7$ are $0,0,0,1,3$. Keeping the exceptional projective index at level two is essential; applying the factor one-half there would give a nonintegral answer. This agrees with the <principal congruence modular curve genus> formula obtained from these branch counts.
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