Homophones are often used humorously in Chinese. Some examples:
Many jokes use the topic of people confusing similar-sounding phrases:
A man selling candy fruit (Tanghulu) was beaten up. At the police office,
the police officer asked the haidresser: "Why did you beat up the
candy fruit seller?" The hairdresser answered: "I was in my shop
applying perms (烫头发)， and he stood outside yelling 'he burned [my
hair] to a paste!' (tang hu lou)!"
There are also more satirical uses of word substitution and homophones. For example, in mainland Chinese online communication, it's common to refer to China's government as the 天朝 (heavenly court), to mockingly compare it to the imperial court of times past. There are many examples of this type.
An example that may be unique to Chinese is the 同音文章 (One syllable article), where a short story is composed only of characters that are pronounced as the same syllable (but not necessarily the same tone). These are often humorous, just because the form constraints enforce absurd content. From a famous example:
The "Stone-house poetry master" from family Shi liked lions
and swore to eat ten lions [...]
There are many other examples from many different types of Chinese humor. In conclusion: Yes, both homophones and words with multiple meanings are used in Chinese humor, both in satire and vulgar humor.