Regarding Question A, the $3$-torus provides a counterexamle: its complex $K_1$-group has rank $4$ but its cohomotopy has only rank $3$. (I have learnt about this example from the book of Rordam, Larsen and Laustsen.)
The diagram in Question B clearly commutes because $K_1(r)$ just acts by picking some representative, restricting it and then taking its $K_1$-class.