If the pre-WW-I flashback scenes are taken at face value, it establishes that this version of Niles Caulder was born circa 1873.
The events in Hair Patrol happen concurrently with the events in Jane Patrol.
Tommy Snider (The Beard Hunter) had to shave multiple times a day while filming to hide his thick beard growth because the character Ernest Franklin, aka The Beard Hunter, has a condition and can not grow any facial hair.
There is a "Beard Hunter" in the comics, but there are some differences: While he does live with his mother, rely on guns, and have a hormone deficiency, he hunts those with beards to kill them and has no superpowers.
The song playing in the grocery is a muzak version of Ambrosia's 1978 top ten hit "How Much I Feel".