You may be interested in arXiv:1311.1460 "On spaces of modular forms spanned by eta-quotients" by Jeremy Rouse and John Webb.
From the abstract:
We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $\,\Gamma_0(N)\,$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level $N$.
Since Eisenstein series can be expressed in terms of Jacobi theta-null functions, and theta-null can be expressed in terms of Dedekind eta functions, you are essentially asking if modular forms can be expressed n terms of Dedekind eta functions. In general,
no, but sometimes, yes.
As for finding eta expressions from a finite $q$-series, I don't know of any algorithms, but it may be possible in some cases. For example, suppose we are given enough terms of the $q$-series expansion of a rational function of
$\, \eta(\tau), \dots, \eta(d\tau), \dots, \eta(N\tau) \,$
where $\, d|N \,$ and given the degree of the rational function. Then it is possible to recover the rational function by solving linear equations using a general algorithm that finds possible algebraic dependencies among finite $q$-series.