From the general theory of modular forms, we have the orthogonal decomposition: $$M_2(\Gamma_1(N))=S_2(\Gamma_1(N))\oplus\mathrm{Eis}_2(\Gamma_1(N)).$$
So for $f\in M_2(\Gamma_1(N))$, we write $f=f_\mathrm{cus}+f_\mathrm{eis}$ accordingly.
$\textbf{My question is:}$ If $f$ has integer coefficients, then is it true that $f_\mathrm{cus}$ and $f_\mathrm{eis}$ also have integer coefficients?
For a generic vector space with a decomposition and a fixed choice of integral submodule, this kind of question is false. But, in this specific case it seems to be true.
No, it's false; and it is false in a way that is interesting enough that people have written hundreds of papers studying exactly how and why it fails. :-)
The fact that the "cuspidal + Eisenstein" decomposition doesn't work integrally is closely related to the fact that there exist nontrivial congruences between cusp forms and Eisenstein series. These Eisenstein congruences are crucial in all sorts of deep research, e.g. the line of work that started with Ribet's converse to Herbrand's theorem and continued via Mazur and Wiles' proof of the Iwasawa main conjecture.