I can't give you your desired "most general" theorem, but I can say a little about this. In (b), the condition "shape(A) is lexicographically larger than shape(B)" is much stronger than it needs to be: "shape(A) is not dominated by shape(B)" will yield the same conclusion (recall the dominance order on partitions: $\lambda$ dominates $\mu$ if $\lambda_1+\cdots+\lambda_i\geqslant\mu_1+\cdots+\mu_i$ for each $i$).

(c) is false as you've stated it – look again at Serganova's notes.