How to understand the reduction from 3-Coloring problem to general $k$-Coloring problem?
The $k$-coloring problem is usually defined only for constant $k$, so $n$-coloring doesn't make sense. For every constant $k \geq 3$, the reduction you mention works. By adding a superconstant number of vertices you can show, for example, that $(n/2+3)$-coloring is NP-complete.