Merge $k$-sorted arrays - without heaps/AVL tree in $O(n\log(k))$?
For the sake of avoiding dealing with rounding in the analysis, assume for simplicity that $k$ is a power of $2$ (if it is not, you can add $d \le k-1$ dummy arrays with one $+\infty$ element each, and ignore the last $d$ elements in the final sorted vector).

You can merge your arrays two by two in a binary-tree fashion: the algorithm works in $\log k$ phases. In phase $i=0, \dots , \log k - 1$ there are $k/2^i$ sorted arrays, where the arrays of phase $0$ are the $k$ input arrays. These arrays are arbitrarily grouped into $k/2^{i+1}$ pairs and each pair is merged. The resulting arrays are moved to the next phase.
At the end of the $(\log k - 1)$-th phase there is only one sorted array $A$ remaining. Return $A$.

The total amount of work on each phase (a level of the tree) is $O(n)$, and the number of phases (i.e., levels) is $O(\log k)$.