Indeed, I think Steven Jay Gould said that the jury is still out on whether intelligence has proved a successful trait for humans, given that we have been around for barely eye blink on evolutionary time scales.

Many, many genes can contribute in some (usually very small) way to depression/addiction/other psychiatric disorders, and a single gene can contribute risk to a variety of disorders. So it's not as clear cut of a thing to identify as whether a gene is involved in the immune system. In fact immune system genes have come up towards the top of genome wide association studies for conditions like Schizophrenia. So it doesn't even have a clean relationship with "genes expressed in the brain".

All that is just to say that there is evidence that some Neanderthal mutations are linked to depression risk, but I wouldn't go so far as to characterize any of the involved genes as depression-related. Maybe that's just semantics though.

More specifically here is a relevant snippet from a study summary [1]: "These analyses supported links to traits such as mood disorders, depression, obesity and different types of keratosis. The most significant associations were for mood disorders and depression, for which Neanderthal SNPs can explain up to ~1% of the risk."

IIRC there was also a (less strong) association with nicotine addiction. Not sure about addiction more generally.

[1] https://www.nature.com/articles/nrg.2016.26