Let $\Gamma\le SL(2,\mathbb{Z})$ be a congruence subgroup of level $N$. Let $R$ be a $\mathbb{Z}[1/N]$-algebra.

Let $\mathcal{Y}(\Gamma)_R$ denote the moduli stack over $R$ of elliptic curves equipped with a $\Gamma$-structure. For an integer $k$, let $\mathcal{Y}(\Gamma,\Omega^k)_R$ denote the moduli stack of triples $(E/S,\alpha,\omega)$, where $f : E\rightarrow S$ is an elliptic curve over an $R$-scheme $S$, $\alpha$ a $\Gamma$-structure on $E/S$, and $\omega$ is a basis of the $\mathcal{O}_S$-module $(f_*\Omega^1_{E/S})^{\otimes k}$.

There is a natural map $p_k : \mathcal{Y}(\Gamma,\Omega^k)_R\rightarrow\mathcal{Y}(\Gamma)_R$ given by forgetting the differential $\omega$. My understanding is that a Katz modular form of weight $k$ is just a section of $p_k$ (perhaps up to 2-isomorphism?). Is this correct? (Here I want to consider weakly holomorphic modular forms - that is, modular forms "holomorphic on the upper half plane, meromorphic at the cusps")

The set of such sections ("weakly holomorphic" Katz modular forms of weight $k$) form an $R$-module $KM^k(\Gamma,R)$. My question is, if we pick an embedding $R\hookrightarrow\mathbb{C}$, any section of $p_k$ can be pulled back to a section of $p_{k,\mathbb{C}} : \mathcal{Y}(\Gamma,\Omega^k)_\mathbb{C}\rightarrow\mathcal{Y}(\Gamma)_{\mathbb{C}}$, and over $\mathbb{C}$, I can more or less see that the $\mathbb{C}$-vector space $KM^k(\Gamma,\mathbb{C})$ is the "same" as the space of weakly holomorphic modular forms for $\Gamma$ defined classically.

My question is - is $KM^k(\Gamma,R)\otimes_R\mathbb{C}\cong KM^k(\Gamma,\mathbb{C})$?

If this is not always true, then under what circumstances will it be true? (perhaps some condition like the Tate curve over $\mathbb{Z}((q))\otimes_\mathbb{Z} R$ admitting "all $\Gamma$-structures"?)

Also, if $\Gamma$ is torsion free, so that $\mathcal{Y}(\Gamma)_R$ is a scheme, then does $p_k$ (over $R$) always admit a section? Equivalently, if $f : E\rightarrow S$ is an elliptic curve equipped with a "fine level structure" (e.g. a $\Gamma(N)$-structure for $N\ge 3$, a $\Gamma_1(N)$-structure for $N\ge 4$,...etc), then is $f_*\Omega^1_{E/S}\cong \mathcal{O}_S$?

I apologize if this is obvious.