  In this paper the authors investigate the $q$-Schur algebras of type B that
were constructed earlier using coideal subalgebras for the quantum group of
type A. The authors present a coordinate algebra type construction that allows
us to realize these $q$-Schur algebras as the duals of the $d$th graded
components of certain graded coalgebras. Under suitable conditions an
isomorphism theorem is proved that demonstrates that the representation theory
reduces to the $q$-Schur algebra of type A. This enables the authors to address
the questions of cellularity, quasi-hereditariness and representation type of
these algebras. Later it is shown that these algebras realize the $1$-faithful
quasi hereditary covers of the Hecke algebras of type B. As a further
consequence, the authors demonstrate that these algebras are Morita equivalent
to Rouquier's finite-dimensional algebras that arise from the category
${\mathcal O}$ for rational Cherednik algebras for the Weyl group of type B. In
particular, we have introduced a Schur-type functor that identifies the type B
Knizhnik-Zamolodchikov functor.
