  In this work, we use monotonicity-based methods for the fractional
Schr\"odinger equation with general potentials $q\in L^\infty(\Omega)$ in a
Lipschitz bounded open set $\Omega\subset \mathbb R^n$ in any dimension $n\in
\mathbb N$. We demonstrate that if-and-only-if monotonicity relations between
potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional
subspace. Based on these if-and-only-if monotonicity relations, we derive a
constructive global uniqueness results for the fractional Calder\'on problem
and its linearized version. We also derive a reconstruction method for unknown
obstacles in a given domain that only requires the background solution of the
fractional Schr\"odinger equation, and we prove uniqueness and Lipschitz
stability from finitely many measurements for potentials lying in an a-priori
known bounded set in a finite dimensional subset of $L^\infty(\Omega)$.
