How to appropriately use the word "baby" as in the pattern " ____ baby ___"; for example, "run baby run"?
As a non-native English speaker, I am not sure what is the meaning of this pattern "_ baby _" as in "run baby run"? Another example: "USA continues to "drill baby drill" (referring to oil). What other verbs can I use for this pattern?
Solution 1:
Baby here is, as Janus Bahs Jacquet says, a vocative naming the person addressed. Baby is ordinarily a term of endearment, but in this context it implies alliance or collegiality rather than affection. This slang use was hip in the 50s and 60s, but it is heard less now.
The repeated verb on the outside is cast in the imperative.
The whole is an eager or urgent exhortation to the person addressed as baby to perform the action of the verb: “Do it, {baby/man/girl/NAME/etc.}, do it!”
Solution 2:
"Baby" here is generally not referring to any particular person. In fact, it's hard to say that it refers to a person at all.
In the 60s culture "baby" was often used as more of an interjection, in some cases approximating "huh?" or "eh?" in meaning, in others perhaps used as you would "guy" or "pal" or "bloke", as in "Whadaya say, baby? Let's go get lunch." (A weak example, but I tried to pick something with minimal innuendo.)
In "Burn, baby, burn!" it's used as an intensifier, giving the rather bland "Burn, burn!" much more punch. Most other uses of the form "word, baby, word" are imitations of the original "Burn, baby, burn!"
Note that most of these uses do not overlap with the affectionate use of "baby" to refer to a child or to one's intimate friend -- that's a different case entirely.
[It is worth noting that, speaking as one who was not there in Watts at the time (and not a card-carrying member of the "counterculture") the phrase "Burn, baby, burn!" achieved notoriety not because it was (necessarily) shouted by everyone in Watts but because some reporters (supposedly) heard it (or perhaps read it in fliers being circulated) and reported it, and the headline "Burn, baby, burn!" then appeared on newspapers across the country. Had the phrase not received that incredible amount of national (and international) publicity at that pivotal moment in history it likely would have vanished without a whimper.
It's also worth noting that the situation that triggered the Watts riots was not substantially different from the ongoing situation in Ferguson, Missouri.]
Solution 3:
It is an intensifier. You can use it informally, though very flexibly. "Eat, baby, eat!" "Drink, baby, drink!" "Win, baby, win!"