  The control of recently observed spintronic effects in
topological-insulator/ferromagnetic-metal (TI/FM) heterostructures is thwarted
by the lack of understanding of band structure and spin texture around their
interfaces. Here we combine density functional theory with Green's function
techniques to obtain the spectral function at any plane passing through atoms
of Bi$_2$Se$_3$ and Co or Cu layers comprising the interface. In contrast to
widely assumed but thinly tested Dirac cone gapped by the proximity exchange
field, we find that the Rashba ferromagnetic model describes the spectral
function on the surface of Bi$_2$Se$_3$ in contact with Co near the Fermi level
$E_F^0$, where circular and snowflake-like constant energy contours coexist
around which spin locks to momentum. The remnant of the Dirac cone is
hybridized with evanescent wave functions injected by metallic layers and
pushed, due to charge transfer from Co or Cu layers, few tenths of eV below
$E_F^0$ for both Bi$_2$Se$_3$/Co and Bi$_2$Se$_3$/Cu interfaces while hosting
distorted helical spin texture wounding around a single circle. These features
explain recent observation [K. Kondou {\em et al.}, Nat. Phys. {\bf 12}, 1027
(2016)] of sensitivity of spin-to-charge conversion signal at TI/Cu interface
to tuning of $E_F^0$. Interestingly, three monolayers of Co adjacent to
Bi$_2$Se$_3$ host spectral functions very different from the bulk metal, as
well as in-plane spin textures signifying the spin-orbit proximity effect. We
predict that out-of-plane tunneling anisotropic magnetoresistance in vertical
heterostructure Cu/Bi$_2$Se$_3$/Co, where current flowing perpendicular to its
interfaces is modulated by rotating magnetization from parallel to orthogonal
to current flow, can serve as a sensitive probe of spin texture residing at
$E_F^0$.
