Likely I am misunderstanding/missing something, but a claim in a paper appears wrong to me.
According to Coloring Graph Powers: Graph Product Bounds and Hardness of Approximation p. 2
Unless $NP=ZPP$, there is no polynomial-time algorithm to approximate $\chi'(L^k(G))$ to within a factor of $n^{1/3-\epsilon}$ for $k \in \{2,3\}$ and $n^{1/2-\epsilon}$ for $k \ge 4$.
$\chi'$ is the chromatic index and $L^k(G)$ is the $k$-th power of the line graph.
Let $\Delta$ be the maximum degree of $\chi'(L^k(G))$. It is easy to compute. The chromatic index is either $\Delta$ or $\Delta+1$. Choosing the latter for the approximation gives absolute error of at most $1$, which is the best absolute error for integers (not counting equality).
This appears to contradict hardness of approximation, since it is the best possible approximation, not counting exact result.
Q1 What is wrong with this seeming contradiction?
I think you are correct. The chromatic index is easy to "approximate".
I believe that the authors meant $\chi(L^k(G))$ instead of $\chi'({L}^k(G))$. Notice that in the Theorem 1 on p2, it is mentioned that the result you state implies the hardness of the strong edge coloring.
On p5, it is told that the strong chromatic index $\chi'_S(G)$ is equal to $\chi(L^2(G))$. Thus the case $k = 2$ is probably what the authors were talking about in Theorem 1, which makes more sense with $\chi(L^k(G))$ instead of $\chi'({L}^k(G))$.