  Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry
of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse
index at most $I$ and total branching order at most $B$. Previous works of
Fischer-Colbrie and Ros explain that such surfaces are precisely the complete
minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total
branching order. Among other things, we derive scale-invariant weak chord-arc
type results for such an $M$ with estimates that are given in terms of $I$ and
$B$. In order to obtain some of our main results for these special surfaces, we
obtain general intrinsic monotonicity of area formulas for $m$-dimensional
submanifolds $\Sigma$ of an $n$-dimensional Riemannian manifold $X$, where
these area estimates depend on the geometry of $X$ and upper bounds on the
lengths of the mean curvature vectors of $\Sigma$. We also describe a family of
complete, finitely branched minimal surfaces in $\mathbb{R}^3$ that are stable
and non-orientable; these examples generalize the classical Henneberg minimal
surface.

Differential Geometry