What benefit does assigning a jailer, farm manager etc have?

What traits, stats, specializations affect those jobs efficiency? and how?

I know bits and bobs about Lab assistant and Headgirl, just missing the whole picture.

Well, Jailers count as a management job for the purposes of Tamer, for one. So having a Tamer as your Jailer means they apply to anyone who's Uncivilized.

Otherwise, the Jailer adds (Wits / 10) Health, (Charm / 8) Obedience, and has a (Confidence - 30)% chance to knock a day off of Captured, to everyone in Jail.