How important is it to use a comma before a coordinating conjunction in writing these days?

The purpose of punctuation is to make it clear how to read the text aloud; sometimes this affects the meaning.

So the comma in your sentence after "Every morning" indicates a pause (with a downward intonation); leaving it out would suggest "I wake up" was a relative clause.

Thus the real issue is whether a reader needs the comma after "6.00 am" in order to read it aloud and easily get the intonation/pause/sense correct.

My answer here would then be that in this simple example omitting the comma does not make it difficult to read aloud correctly straight off; however, with longer sentences of the sort common in academic writing, a comma would be very helpful and necessary.

In general media, people today often seem to omit commas wherever possible; however I would not do this in academic writing where readers may be more pedantic

Just to make it clear: when reading slowly many people are actually "reading aloud in their head"; (if not, then one is technically "skimming"); either way if one cannot read a text out loud correctly first time, then it is not clear; and commas may be needed to make it so. Thus:

"In general media, people today often seem to omit commas wherever possible." is not the same as:
"In general, media people today often seem to omit commas wherever possible."

In short, put commas where needed to make the meaning clear first time so the reader does not have to backtrack and reconstrue the sentence. Elsewhere, commas are functionally redundant.

I'm more concerned about your last sentence: 'How serious is it to mandate that my international include this comma?'

In academic writing (especially, I think, in the US), commas seem to be expected, unless the second clause is very short. But I can't imagine your example occurring in any formal context, so I don't see any problem. In non-academic writing I go with jayles and use a comma when I would pause, rather than worrying about formal rules.

My usage bible, Practical English Usage, and Oxford dictionaries online seem to suggest that commas are only necessary in complex sentences or where clauses are longer:

I came home and the others went dancing.

I decided to come home earlier than I had planned, and the others spent the evening at the local disco.

In academic writing (especially, I think, in the US), commas seem to be expected, unless the second clause is very short. But I can't imagine your example occurring in any formal context, so I don't see any problem. In non-academic writing I go with jayles and use a comma when I would pause, rather than worrying about formal rules.

My usage bible, Practical English Usage, and Oxford dictionaries online seem to suggest that commas are only necessary in complex sentences or where clauses are longer:

I came home and the others went dancing.

I decided to come home earlier than I had planned, and the others spent the evening at the local disco.

Sorry about that. I tapped twice.

Obviously beginners need guidance but it would be sad if punctuation was taught as a series of unbreakable rules. It is so much a question of taste and function.

The important thing to grasp is that punctuation converts to form part of an audio file in your head: listen to the voice in your head as you read this. So the question becomes:
how does one pronounce a comma?
Usually as a small pause with a rising or wavy intonation.

Most text layout affects how the text sounds in your head: consider
poetry and
the effect
of an end-of-line
on intonation
and rhythm.

It is worth noting that graphics such as bar graphs, and pie charts do not produce an audio file in your head; but columns of figures on a spreadsheet or accounting report are usually "read" as audio input.

You will see Americans abuse the language in MANY ways, but should you teach students that this is okay? Of course not.

When two complete sentences are joined into one, it should be done with a semicolon (rarely) or a comma and coordinating conjunction. An exception can be made when both of the sentences are very short and the meaning is clear.

Most international students need written English for business or academic purposes; what is acceptable means what is acceptable in those contexts. This means what is in the writing guides at uni, or roughly what one can find in a quality newspaper or magazine, or relative style guide.

What has happened since I went to school is that punctuation is now often more minimalist, so the original question is a good one. The only realy answer is to punctuate wherever needed for clarity.

There is a similar issue with "But" at the beginning of a sentence: common enough in newspaper articles; but unwise in an English exam, as this is traditionally a no-no.