Learn English – Why are “software”, “advice”, and “information” uncountable

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.

Often the countable and uncountable versions of an English noun will refer to different things. For example, "hamburger". If you say,

I love hamburger

it means that you enjoy the actual ground meat, in all its various uses. On the other hand,

I love hamburgers

means you like the specific use of ground beef in a hamburger, with the bun, lettuce, pickles, tomato, etc. You can also say:

I love a good hamburger

to refer to the specific object, with the relevant qualifiers.

In a similar way:

I love chicken (the meat)
I love chickens (the animal)

I love television (the programs on television)
I love televisions (the electronic device)

These distinctions seem idiomatic and can only be learned by exposure, memorization, and practice.

However (in general) when a noun's countable form and uncountable form both refer to (more or less) the same thing, it seems you use the uncountable form to refer to the generic or overarching concept, and you use the countable form to refer to specific instances or examples of that thing. With cake:

I love cake (cake in general)
I love the cakes Mary served at her party (those specific cakes)

I love coffee (coffee in general)
I love the coffees from Brazil (the various varieties of coffee from Brazil)

Note also a similar use when talking about wishes or desires:

I'd love some cake.
I'd love a cake.

As with anything in English, there may be exceptions.