Do I understand correctly that $G_0$ is just a finite dimensional Lie algebra?

I don't know if this is helpful, but when $n=2$, $U_1$ can be written in terms of $U_0$, so we can regard the algebra as a quotient of a polynomial algebra: $k[x]/\langle x(x^3-4x^2+2x-1)\rangle$. Perhaps it would be easier for someone to recognize this algebra.

How do I interpret your relation when $j=1$ or $n$?

Doesn't setting $x_j=-1$ just change the coefficient of $e_{k-1}$ to $-1$ (same for $e_{n-1}$. In that case, setting one $x_j=-1$ gives you -1 in (1).

I don't think this answers your question, so I'll leave it as a comment. In Khovanov-Lauda's paper `A diagramatic approach to categorification of quantum groups II', equations (11)-(13) give some formulas for the interaction between $T_w$, $w$ the Coxeter element, and certain idempotents acting on polynomials ($W$ is the symmetric group). These formulas are then used to prove the categorical Serre relations, which takes the form of an exact sequence. One of the maps is given by multiplying by the Coxeter element in the quiver Hecke algebra.

Thanks, Peter. I guess I don't really care if there are restrictions on the order, but this certainly answers my question. I would hardly call your proof elementary, though.

Its the bar involution: $\overline{q}=q^{-1}$ and $\overline{E_i}=E_i$ for the Chevalley generators.

You can get Demazure operators by specializing $q=0$ in (an appropriate normalization/integral form of) the Iwahori-Hecke algebra. I believe this is pursued in some form or another in a recent preprint of Dan Bump and collaborators.

I should add that even if the lattices are the same, one can still make sense of all this. For example, the C basis may correspond to simple objects and the C' basis corresponds to so-called standard objects.

How are the $\mathbb{Z}[q,q^{-1}]$-lattices spanned by the C and C' bases related? Often in categorification one sees two lattices. One lattice is spanned by projective objects while the other is spanned by simples. The minus signs in the change of basis can interpreted as taking the Euler characteristic of a projective resolution.