An utmost importance is not natural English. It would not surprise me if the author of your example is not a native speaker.
The issue of English having count nouns and mass (or non-count) nouns is a poor way to talk about nouns in English.
A better way is to say that nouns can be used as count nouns, non-count nouns, or both.
Some nouns in English can be used as both count nouns and non-count nouns. Let's look at freedom.
It can be used as a count noun:
The USA guarantees its citizens many freedoms that other countries do not.
It can also be used as a non-count or mass noun:
Freedom, just like life and liberty, is highly-cherished by most citizens.
Ultimately, what decides whether a noun can be used as count, non-count, or both is the community of English speakers. Some nouns used only as mass nouns today were once used as both.
But importance is used only as a non-count noun in today's English.
As for your actual question:
You cannot use the indefinite article with a singular noun used as a non-count noun.
The following is ungrammatical:
An importance has come to mind today.
You can say:
Waiter, I'll have a cappuccino to drink.
because a cappuccino is conceptualized as a serving of cappuccino.
You can say:
This coffee is a coffee that is out of this world.
because a coffee is conceptualized as a type of coffee.
You can use the definite article with singular or plural count and non-count nouns.
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
Both are possible. Said that, it depends on the context.
Collective nouns are tricky in that way. The example of 'team' is good one to ponder upon.
Say, you are considering the team as one entity. You then use a singular verb 'is'.
But then, if you want to treat each team member separately but still want to use the collective noun, it takes 'are'. Say -
Similarly, India have won the match is also correct.