Homonyms in the context of humor or satire Because my study of Chinese is mostly based on intermediate videos I think I am missing something. Now that I have a sense of the large number of homophones in Chinese, the potential for double-entendres and word play seems obvious; but it is not the sort of thing one would put into intermediate material.


  I wonder if there are some examples of humorous or satirical ambiguity that convey a sense of how these commonly come up?


This can happen in English when a word has two distinct meanings. For example, in an agricultural context we speak of a bull "servicing" a cow; this usage injects an element of humor into the notion that a phone company "services" a customer.
Homophones are often used humorously in Chinese. Some examples:

Many jokes use the topic of people confusing similar-sounding phrases:


  把一个卖糖葫芦的被揍了，到警察局，警察问理发师：“你为什么揍卖糖葫芦的？”理发师说：“我在屋里烫头发，他在外面喊‘烫糊喽’！”
  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:


  石室詩士施氏，嗜獅，誓食十獅。
  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.