I can see you've done a lot of research, and have come up with an ambiguous understanding which is a shame (English sucks). I hope I can help clear the air for you. :)
Your first instinct about there being no article needed in the first example was correct:
Without air and water, living things could not survive.
In fact, in most of the examples your sister found to support the being the correct particle, the could actually be removed entirely:
Birds and insects could not fly without air to support them. Without air, humans would not be able to fly either.
We can’t survive for more than a few minutes without air, so why isn’t air as much a part of us as our legs or arms?
To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without water the protein will never get to the muscles and therefore never get repaired.
In all of the examples above, "air" is being referred to more as a concept than as a tangible thing. We're not talking about a "specific air" that we could hold or touch or see. Also, it's not just a concept of one thing, but "some amount" of it. The sentences above aren't referring to one air, but rather an amount of air. However, like you said, it is uncountable, neither plural nor singular. I'll try replacing the word air with information. Again, we're not referring to any specific information, but rather the concept of some amount of information. I know, it's silly, and doesn't make sense...but it works, grammatically:
Birds and insects could not fly without information to support them. Without information, humans would not be able to fly either.
We can’t survive for more than a few minutes without information, so why isn't information as much a part of us as our legs or arms?
To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without information the protein will never get to the muscles and therefore never get repaired.
See? It works, even if it's nonsense.
In this example, things are a little different:
The sun, the moon, the sea, the sky, the Arctic Circle, the environment, the capital, the air, the ground, etc.
The reasoning is correct, that "the definite article is used in front of things generally regarded as unique." There is only one air being referred to here: the air on Earth.
So to sum up, think of the first example again. Is the sentence referring to an amount of non-specific air? Yes. So, we don't need to use the.
Best Answer
I suppose if you wanted to say something about the "20s decade" of any century, you could say, "I love 20s" or "20s are periods of change" or whatever. But it's such an unlikely thing to say, I'd be surprised if you could find an example.
But to take a more likely analogy, suppose you were talking about the basement of a building. If you were talking about the basement of one particular building, you might say, "I clean the basement." You use "the" because you are talking about one particular basement. But if you wanted to talk about the basement of any building, you could say, "I clean basements." Now there's no article because we are talking about basements in general and not one particular basement.
As "the 20s" would normally be understood to mean "the 1920s", we are talking about one particular period of time, so you use the singular and the definite article. (I suppose if someone in 1840 talked about "the 20s", he presumably meant "the 1820s", and if 30 years from now people talk about "the 20s", they'll probably mean "the 2020s", unless the context makes clear that you're talking about some other century.)
Note that regardless of whether you are talking about a specific thing or a general thing, if you use the singular, you need an article, a possessive, the number "one", or one of a few other special pronouns. Like if we are searching for your lost dog, I might say, "I see the dog" or "I see your dog." If we are wandering about and see an unknown dog, I would say, "I see a dog." It would never be correct to say, "I see dog." (Well, you could say "I see Dog" if you are talking about Dog the Bounty Hunter, a person named "Dog". But then it's a proper noun.)
As Mick notes in his comment, you could use "20s" without an article if you were using it as an adjective rather than a noun, and the noun in question does not call for an article. Which would usually mean that it is either plural or uncountable. "I like 20s music" (uncountable) or "There were several 20s cars in the antique car show" (plural).