The usual, humorous, phrase is like this:
My daughter is 16, going on 32!
It simply means she is precocious. She is only 16, but she already behaves in a very mature way. So in the usual phrase, the age difference goes upwards. Another example, "My kindergarten kid is 5 going on 10!"
That's the usual direction of the joke. But here, the author here is deliberately turning it around the other way.
So indeed the author is expressing that the woman is 22 but behaves immaturely.
To understand this usage:
In English commercial writing, in the present day, there is a fad to take an existing humorous phrase, and "turn it around". The idea is that it (supposedly) sounds even more witty when reversed. You could say this is an "overused trick" in English commercial writing today. The example at hand is precisely an example of that process.
(Note: as Robusto explains, "going on" very simply means "almost". For example, "to walk to the store is five, going on six, miles", "renovation costs are 80 thousand, going on 90 thousand.")
So, to get the entire feel of the passage in English relies on the following chain:
1) "Going on" means "almost": the child is six going on seven. That sentence simply means "almost seven".
2) Very commonplace humorous use of "going on" with a large gap going upwards, used specifically of precocious children: that girl is 15 going on 35!
3) In this case, the author has "turned around" that usual humorous pattern: "the person is 35 going on 15". Note again that it is common (today) in commercial English to invert a common humorous construction, to create a (supposedly) even funnier one.
By the way, the phrase Sixteen Going on Seventeen is indeed one of the handful of most famous "showtunes" in all of English, 1965,
https://www.youtube.com/watch?v=hwK_WOXjfc0
(immortal performance in the film by Charmian Carr) So that song was written by Oscar Hammerstein II and Richard Rodgers (the two most famous "showtunes" era composers) and it's one of the songs in The Sound of Music (far and away the most popular musical show and film in English).
So, for any English speaker, whenever you say or hear the phrase "16 going on 17" (much as with "do-a-deer", "brown paper packages" "edelweiss" and indeed others from the same show) it associates instantly with the song.
The phrase stir the paint is not standard slang. It cannot be found in this UK dictionary of slang, this slang site, this other slang dictionary, or this published dictionary of slang. However, there are similar phrases which have sexual meanings. According to the Cassell's Dictionary of Slang, these are some of the slang uses of stir as a verb:
stir fudge--1960s, to perform anal intercourse
stir it up--1950s, to masturbate
stir one's stew, stir the batter, stir the sauce--1950s+, see stir it up
stir shit--go out of one's way to make trouble
stir the porridge--1980s, Australia, to have sexual intercourse with a woman immediately after she has had intercourse with another man, esp. used of the final man in a gang-rape
stir the possum--to create a disturbance
stir the stew--1900-1910s, to have sexual intercourse
So there is a highly sexual connotation to stir that has been around since the 1900s. It seems likely that stir the paint is similarly sexual, although it is not documented before the Family Guy episode aired. Thus it is also likely that Family Guy coined the phrase, but did so based on these preexisting meanings. I would hazard a guess that the "paint" is menstrual blood.
It is similarly possible that stir the paint is not sexual at all, but instead refers to some dynamic between the two parents in which Lois will not let Peter do tiny things like stir paint, but given the nature of the humor in Family Guy I doubt this.
Best Answer
It's not strictly literal, the subject may in fact have some ideas. To "have no idea" means that whatever ideas the subject does have, if any, are either highly uncertain (usually, when the speaker is the subject) or presumed to be incorrect (usually, when the subject is someone else).
If I say "I/we have no idea", I mean that I am / we are highly unsure, though I probably have at least some vague thoughts on the matter. If I say "he has no idea", I mean that I think he is either ignorant or wrong.