TheFreeDictionary could be clearer; essentially, what it means is something more like the Wiktionary definition:
give a damn (third-person singular simple present gives a damn, present participle giving a damn, simple past gave a damn, past participle given a damn)
To be concerned about, have an interest in, to care (about something).
Frankly, my dear, I don't give a damn.
I'd question the assertion that it's always used in the negative; the Wiktionary entry lists a non-negative construction, and so others surely exist, such as your example.
To TRomano's point about the confusion: I believe the phonologically reduced n't in couldn't causes the expression's meaning to be carried in the intonation rather than the lexicogrammar.
That is, because the n't is so difficult to hear sometimes, the following sentences can have the same meaning, depending on the tone:
I couldn't care less
I could care less
Having said that, I'm not sure the change is complete for give a damn, or even that it's necessarily the same thing. I'd want to see more examples of them.
For instance, COCA provides the following results:
(A) -not/n't GIVE a damn 252
(B) not/n't GIVE a damn 547
(C) not/n't * GIVE a damn 24
Note that in the above results, *
means one word and not any number of characters as it common RE.1 Additionally, GIVE means the lemma - all forms of - give (given, giving, gives)
Firstly, what this shows - even if there could be distant, non-negative constructions - is that overwhelmingly it is used in the negative, but not always.
Consider the excerpt from (A) (above, apologies for the small text here's a direct link), in particular the ones without negation; lines 3 and 15. These examples don't have any distant or proximal negation, and their meanings are clear.
For (3), it is we are smart enough to care. For (15), it is How am I going to care about selling someone a T-1 line?
Of course, language changes, and spoken language so much more rapidly than written language. If give a damn adopts the same meaning whether it is negated or not, it is still in the process of doing so.
Also note that having a negative connotation is not the same as being negated; things can have a negative connotation ("have bad axiological relations") and not be negated. And, looking at the corpus data for hoot and shit (links below), the same is true for them; they retain both a negated and non-negated meaning.
She does not give a damn about her job = She gives a damn about her job
tl;dr: No, at the very least, the above clauses are not equivalent yet, and I'm not sure they're in the process of becoming equivalent. This applies to hoot and shit as well.
Here are links for each of the searches:
The phrase and all that is strictly informal, because it’s sloppy.
A person says and all that when they know that greater precision is called for but they don’t want to make the effort to find words to describe the category they’re talking about. It’s common and ordinary in everyday, informal speech.
It’s grammatically correct even without a word after that. That functions as a demonstrative pronoun, and all modifies it to mean that you’re regarding that as a whole somewhat carelessly, without regard to details and distinctions.
An informal phrase in a formal context
One reason you find and all that in books is because it occurs in dialogue. Books are normally written in formal English, but they can contain informal English in quotations. In indirect speech, appending and all that can suggest that someone is taking a sloppy, dismissive attitude toward a topic. (In informal contexts, the phrase doesn’t carry the connotation of dismissiveness nearly as often.)
Occasionally, deliberate informality in a formal context makes good rhetoric. For example, there is a textbook on vector calculus titled Div, Grad, Curl, and All That. The title, by using the phrase and all that, suggests that the book will cover the topic informally, in a tone and style that are easier to understand than the very formal approaches more commonly found in math textbooks.
The phrase and all that occasionally appears in book titles or other formal contexts to allude to a parody of English history, 1066 and All That by Sellar and Yeatman, published in 1930. Its premise is that it presents English history as you actually remember it from school. For humor, it mixes things up and reduces people and events to absurd simplicity. For example, the book incorrectly calls Alfred the Great “Alfred the Cake” because of a famous story about the real Alfred and some cakes. It rates nearly every king “a good king” or “a bad king”. The book is well-known enough that people sometimes write and all that to designate an irreverent or false history of another topic, either a deliberate one or the way people have misremembered it.
Formal equivalents
There are equivalent phrases in formal English that also avoid explicitly describing some category, leaving the reader to fill in the details: and the like, and such, and similar things. If you can replace things with a more-precise noun, that’s better in formal English. Another formal choice is et cetera, Latin for “and the rest”, usually abbreviated etc.
You could do a lot with this item, such as writing, drawing, and the like.
You can do a lot with this item, including writing, drawing, and the like.
With this item, you can do writing, drawing, and the like.
I removed your first like to avoid repeating the word.
In this particular example, it’s probably better to avoid the phrase entirely, since like, such as, and including all clearly indicate that the list is not exhaustive:
You can do a lot with this item, including writing and drawing.
That has the clarity and crispness of formal English.
Best Answer
I think in this context "a rip" is simply a euphemism, a more socially acceptable version of a stronger swear word. One can, after all, give or not give:
-a damn
-a darn
-a good goddamn
-a fuck (possibly even a flying fuck)
-a shit (or two shits)
-a hoot (or even a hoot in hell)
-a tinker's damn/dam
-a pair of dingo's kidneys
or many, many other things. Many of them would probably not be said by a pastor giving a sermon! So to give the same sense, but in a less offensive manner, you can choose another word, and "a rip" happens to be one of the idiomatic choices that appears to stem from a general association of "rip" with worthlessness.