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}?

[online code](https://sagecell.sagemath.org/?z=eJxVjkEOwiAQRfck3IElkLYWEjc23MKdcYG2WpICIwwLby-JmNTdvPf_T8Z5iAmZB29xpQTSYo7jOH158DDMkE21O1MJ8W3OqSyU5KJMi0Iunm_W32bLwokx3ishZZAt3uKTB9FddNeEC4-roAS12TXUQf8G-ZWQa_GH7QYn6rLBfY3A6x89avEB80BGGA==&lang=sage&interacts=eJyLjgUAARUAuQ==)

**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.