Learn English – “Thank you for the good advice” – can this refer to a single piece of advice

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.

It's all depends on 'context'.

Sometimes we use "have a good one" for a single day (good day)

and sometimes for a longer duration. (Have a good one" [trip])

Let's say you tell me you're going on vacation next week. I say oh wow I'm not going to see you until the end of January. When we finish talking, I say to you "have a good one" I'm telling you to have a good vacation because I won't see you until after you have returned. [think of until we meet again]

A really common use for it is for people who don't know each other very well. They don't know if they'll ever see the person again (but they don't want to say that) so they say "have a good one" (with whatever it is you're going to be doing after I leave you).

So, if I just meet you and we had a short discussion on grammar and it's just ended I'm going to say to you "have a good one" [I don't know you and I don't know what you're going to do after because we didn't talk about that. So it's like saying okay, good luck and enjoy what it is that you're going to be doing now.]

In reference to your question: Most of the time it is used for shorter periods of time but it can also be used for longer periods of time. But it all depends on 'context'.