Uncountable Nouns – Is the Sentence ‘Paul Drinks Much Milk’ Grammatically Correct?

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.

In your example, you could use pieces, as in I have thirty pieces of homework to grade every week.

piece noun [ C ] (THING)
a single object of a particular type:
a piece of furniture/clothing/equipment
a piece of paper (= a whole sheet)
a piece of china (= an object made of china)
a piece of information/advice
(Cambridge Dictionary)

However, that doesn't seem particularly idiomatic to me. You could use assignments, as in homework assignments:

assignment
noun [ C/U ]
us ​ /əˈsɑɪn·mənt/
a particular job or responsibility given to you:
[C] The homework assignment was to read Chapter 2 in our history book.
(Cambridge Dictionary)

However, in my experience, it's more common to use the type of assignment instead of homework. I think the most broad term is assignment, but you could be more specific:
I have 30 ______ to grade every week.

• assignments
• papers
• essays
• worksheets
• modules
• warm-ups
• tests
• quizzes
• etc.

Edit:

I was very briefly a grader (or, "reader") in a related field. I can't remember exactly how I talked about it, but if someone asked me, "How much homework do you need to grade?", I would probably reply

I need to grade thirty [assignments] every week.

You could also say sets (as others have mentioned), or even submissions (more generic). I'm thinking maybe even "papers", but that's usually used with reports or essay-like works.

I don't think I would have responded in the form you supplied, "I need to grade thirty (units) homework every week." But, that's just my personal feeling of it. You can still use pieces, as mentioned earlier. It may or may not sound slightly strange to the listener, but you will be understood.

To my surprise, BrE users are reporting that pieces of homework is idiomatic to them. I did a little Ngram search, and it appears that the phrase is more common in BrE.

I'm from the West Coast (US).