Blockwise determinantal ideals are those generated by the union of all the
minors of specified sizes in certain blocks of a generic matrix, and they are
the natural generalization of many existing determinantal ideals like the
Schubert and ladder ones. In this paper we establish several criteria to verify
whether the Gr\"obner bases of blockwise determinantal ideals with respect to
(anti-)diagonal term orders are minimal or reduced. In particular, for Schubert
determinantal ideals, while all the elusive minors form the reduced Gr\"obner
bases when the defining permutations are vexillary, in the non-vexillary case
we derive an explicit formula for computing the reduced Gr\"obner basis from
elusive minors which avoids all algebraic operations. The fundamental
properties of being normal and strong for W-characteristic sets and
characteristic pairs, which are heavily connected to the reduced Gr\"obner
bases, of Schubert determinantal ideals are also proven