My question is: Does there exist a nonzero module over a non-Noetherian ring with no nonzero finitely presented submodule?
For any element $m$ of a left (right) $R$-module $M$, the submodule $Rm$ ($mR$) is finitely generated, but not finitely presented unless $Ann(m)$ is a finitely generated left (right) ideal of $R$.
Also, while a ring can have a non-finitely presented principal ideal (e.g. the principal ideal generated by $x_1 + (x_1x_2, x_1x_3, x_1x_4, ...)$ in $\mathbb{Z}[(x_n)_{n \ge 1}]/(x_1x_2, x_1x_3, x_1x_4, ...)$), any ring is obviously a finitely presented module over itself, so modules that answer the question must be different from the base ring.
Sure. For instance, let $R$ be any ring with a maximal (left) ideal $I$ which is not finitely generated, and let $M=R/I$. Then the only nonzero submodule of $M$ is $M$ itself, which is not finitely presented.