This word order pattern looks like inversion but it is not. It's a device known as fronting. There are many kinds of fronting, of which this is one. It is possible to "front" the verb or verb phrase, with the verb usually in the base form, followed by the grammatical subject and with an auxiliary to complete it, as in Gisele's examples. There is no inversion of the grammatical subject and the verb; only the bare infinitive part of the verb precedes the grammatical subject, while the auxiliary verb, which carries the tense and number information, always follows it.
Of course the usual versions of these sentences do not repeat the verb phrase. For example, the first sentence would normally be
They advised me to stay, and I did [so]
Notice that the entire verb phrase (verb plus complement) can be fronted:
...and give a party I will
It is also possible to front adjectives:
He was precocious as a child, and precocious he still is
They said it would be cheaper to go by air, but cheaper it certainly wasn't
...and prepositional phrases:
He lived with a chip on his shoulder and with a chip on his shoulder he died
Fronting is a feature of "information packaging," which arranges the parts of an utterance so as to create a desired information flow. It is used to preserve the order of ideas, "old" information preceding "new" information.
Fronting is seen in utterances such as
We remodeled the kitchen in 1998. The bathrooms we left until 2000
He's very charming. Her I can't stand
The kind of fronting Gisele presents is used to create a special effect, often for emphasis, contrast, or sometimes irony.
If there is no special reason to front the verb complement, the fronting is not appropriate. One would have to see the sentence in question to know whether it is a candidate for this special treatment.
Marilyn Martin