Yes, this tends to be more common in informal English and all this makes perfect sense. Verbs (it doesn't have to be just auxiliary verbs) that come at the beginning of a sentence are often left out in spoken English (or in situations where spoken English is being imitated such as chatting over an instant messaging app) for the sake of brevity. One reason this is possible is because from the context it's still quite clear what you are saying (no essential information is lost, in other words):
(Are) you married?
(Are) you done?
In the last example (the paragraph written in informal English), written is something called a past participle. When past participles are used like that, you can always paraphrase the sentence by inserting that is or that are between the noun and the past participle:
the paragraph that is written in informal English
This is normal for past participles. It has nothing to do with the level of formality you're speaking with.
I'm not sure about example #2. Where is there supposed to be an auxiliary verb that has been left out of the sentence?