  In the former part, we study the gravitational collapse of pressureless dust
and find special solutions, where, in both the physical and fiducial sectors,
the exterior and interior spacetime geometries are given by the Schwarzschild
spacetimes and the Friedmann-Lema\^itre-Robertson-Walker universes dominated by
pressureless dust, respectively, with specific time slicings. In the case where
the Lagrange multipliers are trivial and have no jump across the matter
interfaces in both the physical and fiducial sectors, the junction conditions
across them remain the same as those in general relativity (GR). For
simplicity, we foliate the interior geometry by homogeneous and isotropic
spacetimes. For a spatially flat interior universe, we foliate the exterior
geometry by a time-independent flat space, while for a spatiallycurved interior
universe, we foliate the exterior geometry by a time-independent space with
deficit solid angle. Despite the rather restrictive choice of foliations, we
find interesting classes of exact solutions that represent gravitational
collapse in MTBG. In the spatially flat case, under a certain tuning of the
initial condition, we find exact solutions of matter collapse in which the two
sectors evolve independently. In the spatially closed case, once the matter
energy densities and the Schwarzschild radii are tuned between the two sectors,
we find exact solutions that correspond to the Oppenheimer-Snyder model in GR.
In the latter part, we study odd-parity perturbations of the Schwarzschild$-$de
Sitter solutions written in the spatially flat coordinates. For the
higher-multipole modes $\ell\geq2$, we find that, in general, the system
reduces to that of four physical modes, where two of them are dynamical and the
remaining two are shadowy, i.e., satisfying only elliptic equations.
