India, like Italy and China, has no single dominant cuisine: Indian food comprises many different styles of cooking, with each a product of their regional influences, from the fiery vegetarian dishes of the south to the Portuguese-influenced Goan cooking of the west, to the more familiar Mogul food of the north.

(A) with each a product of their
(B) with each as a product of its
(C) each products of their
(D) each a product of
(E) each products of

Could someone explain what type of modifier it is when we use both or each ? And the role it plays . And what is the difference between options B and D ?

The Acoma and Hopi are probably the two oldest surviving Pueblo communities, both
dating back at least a thousand years.
A. both dating
B. both of which have dated
C. and each has dated
D. and each one dating
E. each one of which date

Almost same as above , that is what kind of modifiers these are ?I am trying to understand wha kind of strategy to follow ?