To use one ye olde example, I'm not well placed for Rome or Ancient Egyptian humor so hopefully others can answer for use: In the three kingdoms era (190-280 AD China), wit could be valued (though not always recorded), jokes could be of a personal nature between two friends, to make a point or to protect one's own dignity, word play. If your lucky, a joke might be kept as an example of someone's wit or becuase it was deemed important due to other events but sometimes just get told the person was a wit or a master of Pure Conversation (a popular way of discussing philosophical theory using wit) without examples of their wit. I'm unaware of any jokes involving genitalia and if there were, they were not recorded but we do get some pranks or crude jokes so they did exist:

In terms of pranks, the future Wei Emperor Cao Pi knew that officer Wang Zhong had once been forced to cannibalism by famines so Cao Pi had followers gather skulls from tombs and tie them to Wang Zhong's horse saddle.

From an earlier time: When the Han officer Tian Hui became distressed that his elder brother Weidu was not getting nominated for office but he was. He became ill and lost his voice, his friends thought he was faking so they stole his bedding during the night as a prank then had to apologize when they realized he hadn't call out in alarm and was indeed mute.

In terms of non pranks, we get one joke recorded from the Shu officer Jian Yong as an example of his style of humor. During a time of drought, his lord Liu Bei banned brewing of alcohol and officials wished to punish a man who had the equipment for it. One day Jian Yong was sharing a carriage with his lord, who had been a friend from childhood and who Jian Yong had followed across the land, when Jian Yong spotted a couple walking past “As those people wish to commit adultery, why not arrest them?” and when Liu Bei asked how his friend could possibly know that Jian Yong replied “They have the tools for it, so they are the same as those wine brewers.” An amused Liu Bei pardoned the alleged brewers.

At an earlier occasion Liu Bei was the guest general of another warlord called Liu Zhang (whose land Liu Bei would later take). At a banquet in Fu Liu Bei spotted the soothsayer Zhang Yu who had a famed beard and using wordplay around Liu Bei's past, decided to call Zhang Yu "Mr hair-covered buttocks". Zhang Yu used the same wordplay and Liu Bei's lack of a beard to call him "Mr Exposed Buttocks". Liu Bei may have started the insults but he never forgave Zhang Yu and would, years later, execute Zhang Yu at least partly for that and partly for a negative prophecy that leaked out.

Sources: Cao Cao's SGZ annotation Weilue, Jian Yong's SGZ, Zhou Qin's SGZ (Zhang Yu's misfortune is in there) translated by Yang Zhengyuan

Rafe De Crespigny A Biographical Dictionary of Later Han to the Three Kingdoms 23–220 AD