Does 3-fold improper rotation axis automatically imply a horizontal plane as a s...

The lower-order improper axes have direct equivalents: $S_1$ is just a plane of symmetry $(\sigma)$ and $S_2$ is a center of inversion $(i).$ By definition, an $S_3$ operation is a $C_3$ rotation followed by a perpendicular horizontal reflection $(\sigma_\mathrm h).$

If you look at it group-theoretically and cube the operation, the net result collapses into a pure horizontal plane of symmetry $(\sigma_\mathrm h)$ since $C_3^3 = E$ and $\sigma_\mathrm h^3 = \sigma_\mathrm h:$

$$(S_3)^3 = (C_3\,\sigma_\mathrm h)^3 = C_3^3\,\sigma_\mathrm h^3 = E\,\sigma_\mathrm h = \sigma_\mathrm h.$$

Does having an $S_3$ axis in a molecule strictly guarantee that $\sigma_\mathrm h$ must exist as an independent symmetry element in the point group, or can an $S_3$ axis exist without a distinct $\sigma_\mathrm h$ present?

Complementary to @IanBush's comment, the freely accessible interactive Symmetry Resources at Otterbein University may help you grasp theory, and provide training.

In the particular instance of improper rotations the site explicitly defines

An improper rotation is performed by rotating the molecule $360°/n$ followed by reflection through the plane perpendicular to the rotation axis. If the resulting configuation is indistinguishable from the original, we say there exists an n-fold improper rotation axis (or $S_n$ axis) in the molecule.

with the example of the staggered conformation of ethane (point group $S_6$):

The illustrations on the site are generated by Jmol which allows for a structure read to compute and assign the symmetry operators and character tables. If you are not comfortable to script (as in computation, in Jmol this requires calulate pointgroup; followed by draw pointgroup; on the program's console, ref), Avogadro2 equally allows to assign and visualize the the operators from a GUI (see an earlier answer here).