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The complete calculation is done in Lusztig's book (Lemma 42.1.2). Essentially, the condition $b\geq a+c$ comes from the Serre relations; e.g. we have $$E_1E_2E_1=E_1^{(2)}E_2+E_2E_1^{(2)}.$$

It is quite possible formulas exist or can be found. I don't know of any off the top of my head, but I know that it amounts to counting semistandard tableaux. I do not know of a formula for this numbe …

It is not true that the crossing is necessarily in degree $1$. More precisely, if $\delta_{k,\mathbf i}$ is the crossing between strands $k$ and $k+1$ for the sequence $\mathbf i=(i_1,\ldots, i_m)$, t …

It's the Hopf-algebraic version of the semidirect product of groups. To see this, just consider the case you have groups $G, H, G\rtimes H$. The group rings (over, say, a field $k$) are Hopf algebras …

Yes, there certainly is a quantum version of the Littlewood-Richardson decomposition (in the generic parameter case) for types $A,B,C,D$ (I don't know about the exceptional types). The generalized Lit …

Here is a possible affirmative answer to question (2). (Which is to say, I am not aware of any non-quantum way to answer this question.) I assume from the original post that it is sufficient motivatio …