Working with precision 500 decimal digits, mpmath in sage computes:
$$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$
We believe the LHS of \eqref{1} diverges, so this isn't true.
Q1. Are there theoretical reasons for mpmath to compute \eqref{1}?
Added Despite the interesting answers, I am ready bet mpmath doesn't do any analytic stuff not related to summation, it works purely numerically and the function is treated as black box, returning real number.