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It is straightforward to see that the additive twists $$ f\left(\tau+\frac{M}{N}\right)=\sum_{n\in \mathbb{N}}a(n)\mathrm{e}^{2\pi \mathrm{i}\frac{M}{N}}\,q^n,\qquad M\in\mathbb{Z},$$ lie in $M(k,\Gamma_1(N^2))$, because $\left(\begin{smallmatrix}1 &\frac{M}{N}\\ 0 & 1\end{smallmatrix}\right)$ normalizes $\Gamma_1(N^2)$. Taking linear combinations of these additive twists, all the forms $$ f_m(\tau)=\sum_{\substack{n\in \mathbb{N}\\n\equiv m\ (N)}}a(n)\,q^n,\qquad m\in\mathbb{Z}, $$ lie in $M(k,\Gamma_1(N^2))$, hence so does your $g(\tau)$.

We can express the form $g(\tau)$ as $$ g(\tau)=\sum_{d\mid N}\mu(d)\sum_{\substack{n\in\mathbb{N}\\d\mid n}} a(n)\,q^n.$$ In the notation of Atkin-Lehner (Hecke operators on $\Gamma_0(m)$, Math. Ann. 185 (1970), 134-160), the inner sum is $(f\mid U_d)\mid B_d$, which lies in $M(k,\Gamma_0(dN))$ by Lemmata 2, 6, 14 in the paper. This implies that $$ g\in M(k,\Gamma_0(N^2)). $$

In general, twisting a modular form by a Dirichlet character results in a modular form. See Theorem 7.4 in Iwaniec: Topics in classical automorphic forms.

It is straightforward to see that the additive twists $$ f\left(\tau+\frac{M}{N}\right)=\sum_{n\in \mathbb{N}}a(n)\mathrm{e}^{2\pi \mathrm{i}\frac{M}{N}}\,q^n,\qquad M\in\mathbb{Z},$$ lie in $M(k,\Gamma_1(N^2))$, because $\left(\begin{smallmatrix}1 &\frac{M}{N}\\ 0 & 1\end{smallmatrix}\right)$ normalizes $\Gamma_1(N^2)$. Taking linear combinations of these additive twists, all the forms $$ f_m(\tau)=\sum_{\substack{n\in \mathbb{N}\\n\equiv m\ (N)}}a(n)\,q^n,\qquad m\in\mathbb{Z}, $$ lie in $M(k,\Gamma_1(N^2))$, hence so does your $g(\tau)$.

In general, twisting a modular form by a Dirichlet character results in a modular form. See Theorem 7.4 in Iwaniec: Topics in classical automorphic forms. Your question is the special case of twisting by the principal character modulo $N$. The quoted theorem shows that your twisted form has level $N$.

In general, twisting a modular form by a Dirichlet character results in a modular form. See Theorem 7.4 in Iwaniec: Topics in classical automorphic forms.