  We investigate the effects of magnetic and nonmagnetic impurities on the
two-dimensional surface states of three-dimensional topological insulators
(TIs). Modeling weak and strong TIs using a generic four-band Hamiltonian,
which allows for a breaking of inversion and time-reversal symmetries and takes
into account random local potentials as well as the Zeeman and orbital effects
of external magnetic fields, we compute the local density of states, the
single-particle spectral function, and the conductance for a (contacted) slab
geometry by numerically exact techniques based on kernel polynomial expansion
and Green's function approaches. We show that bulk disorder refills the
suface-state Dirac gap induced by a homogeneous magnetic field with states,
whereas orbital (Peierls-phase) disorder perserves the gap feature. The former
effect is more pronounced in weak TIs than in strong TIs. At moderate
randomness, disorder-induced conducting channels appear in the surface layer,
promoting diffusive metallicity. Random Zeeman fields rapidly destroy any
conducting surface states. Imprinting quantum dots on a TI's surface, we
demonstrate that carrier transport can be easily tuned by varying the gate
voltage, even to the point where quasi-bound dot states may appear.
