He should not show up on unemployment numbers since he retired. His job would likely be counted as a “new” job when he went back to work. This was common in our state government until an investigative reporter did several stories on several people that retired and went back to work in their old job. When people found out it was happening they raised a stink and put an end to it here. The feeling is that if they want to go back to work after retirement they need to take an entry level job and not go right back into their old position. Most declined to do that I am pretty sure.
It doesn't, at least not directly. Indirectly, it increases government's parasitic load on the economy, thus making it harder for employers to employ.
If he had really retired, and the county had hired one of those recent grads living in their parents' basement, that would have reduced the unemployment number. Unless, of course, the retiree had gone looking for another job unsuccessfully.
But the real question here is, how long does your neighbor have to work in his new old job before he can retire a second time?