  For mobility exponents $n \in (2,3)$, we prove existence of strictly positive
solutions to stochastic thin-film equations with singular effective interface
potential and Stratonovich-type lower-order terms. With the perspective of
using these solutions in Part II to construct surface-tension-energy
dissipating solutions to stochastic thin-film equations with compactly
supported initial data, for which finite speed of propagation is shown in
future work, we establish decay estimates on the sum of surface-tension energy
and effective interface potential -- without relying on further functionals
involving initial data. Besides an extension of earlier techniques used in the
case $n=2$ and a refinement of oscillation estimates for discrete solutions,
the main analytical novelty of this paper is a discretization method which
shows nonnegativity for a finite-element counterpart of the integral
$\int_{\mathcal O} (u^{n-2}u_{x})_x u_{xx} dx$ under periodic boundary
conditions in the parameter regime $n \in (2,3)$. This nonnegativity property
serves to control It\^o-correction terms in the estimate for the decay of the
surface-tension energy. This way, it is the key to obtain the desired decay
estimates for the sum of surface-tension energy and effective interface
potential which permit to establish the singular limit of vanishing effective
interface potential in Part II.
