For the first question, I would use the dual pairing with $U_q(\mathfrak{sl}_N)$. The $u_i^j$'s are defined to be matrix coefficients of the vector representation of $U_q(\mathfrak{sl}_N)$ with respect to some distinguished basis, usually a basis of weight vectors. There are unfortunately a lot of different conventions in use. My standard reference is Klimyk and Schmudgen. See, for example, Theorem 19 of Chapter 9 of their book. It states:

There is a unique dual pairing $( , )$ of Hopf algebras between $U_q^{ext}(\mathfrak{sl}_N)$ and $\mathcal{O}(SL_q(N))$ such that $(f, u^k_l) = t_{kl}(f)$ for all $f \in U_q^{ext} (\mathfrak{sl}_{N})$.

Here $((t_{kl}(f))$ is the matrix for $f$ in the vector representation. OK, this theorem is a little bogus in the sense that it is more of a definition. But the point is that $\mathcal{O}(SL_q(N))$ is generated by the matrix coefficients of all finite-dimensional irreducible representations of $U_q^{ext} (\mathfrak{sl}_{N})$, and these separate points of $U_q^{ext} (\mathfrak{sl}_{N})$, so the pairing is nondegenerate.

So, to show that your two guys are equal, just show that they pair the same way with $U_q^{ext} (\mathfrak{sl}_{N})$. Since it is a pairing of Hopf algebras, you just need to check on the generators $E_i, F_i, K_\lambda$. This just requires you to have a handle on the vector representation. In my opinion this is much cleaner than doing the calculations directly.