Learn English – “RATHER THAN forgive AND forget” what does it mean

In this context, you're almost right in that she's saying they don't care for them anymore. But it seems to me that what she's getting at is, they aren't even pretending to care, hence sending someone who is so obviously unqualified to do a job.

Brace yourself now, because things are about to get literal:

"You see, they're just pissing on us without even giving us the courtesy of calling it rain."

Imagine that you're walking down the street, minding your own business, when all of the sudden you find yourself sprinkled upon by some wet substance that smells like pee. After hustling out of the way (how gross!), you look up and see an old man peeing off of a balcony. He's totally oblivious to the fact that he just peed on you and finishes up, and then after a few moments looks down at you and realizes what he'd just done. He quickly looks around to see if there's something else that he could place the blame on, and eventually settles on blaming the weather. "Well now, it looks like we've got a bit of rain today..." he calls down, and then hurriedly rushes out of sight, realizing that he'd just done something totally wrong.

Although lying about it wasn't courteous at all (calling it so would be a grand example of sarcasm), let us contrast it with the following situation:

The next day, everything above happens again down a different street and with a different old man. But this time, after he finishes up, he looks down at you. You glare at him, expecting maybe an explanation or an apology. Instead, he locks eyes with you, grunts, and shakes a little bit more out before sitting down and reading his newspaper. You see, he's just pissing on you without even giving you the courtesy of calling it rain. In this context, hopefully the meaning is more clear.

A more common use of "give/show sb. the courtesy" would be:

You could have shown us the courtesy of letting us try to fix it before giving us a bad review on Yelp.

This could be a business's reply to a customer's negative review on yelp, if the business owner felt that it was an unfair review. A more graphic (and somewhat common usage) is as follows (apologies for vulgarity if it offends):

If you're going to fuck me, you could at least show me the courtesy of buying me dinner first.

(This is a play on words -- "to fuck" can mean both "to have sex with" and "to wrong someone/to mistreat someone.") This is something that a person who feels betrayed might say to their betrayer. E.G. disgruntled employee to boss, spouse A to spouse B upon being served with divorce papers without warning, etc... Again, that last one is pretty vulgar so I wouldn't use it in day-to-day speech, but it serves the purpose of showing how this phrase is used.

"Classical Mechanics is a well known field of Mathematics." Do you even listen to yourself...? – Najib Idrissi 19 hours ago

The user "Najib Idrissi" asks the author of the sentence Classical Mechanics is a well known field of Mathematics. to read that statement for themselves. Najib makes sure that the author will realize the scientific mistake in that sentence and, in short, "will come to their senses."

It's like your friend has told you something very oddly wrong, and you didn't expect him to say so. You tell him/her to listen to himself/herself, so that s/he will realize the mistake they've made. In this case, it seems very rudimentary to the mathematicians that "classical mechanics" are a physics' subject, and hardly related to math. So, "well known", as it means that "many are familiar with it being a part of mathematics" seems a very idiotic expression to them, as if the author didn't know what they were typing, or simply, talking about.

If you ask me, there is a bit of negative connotation in this sentence, but it wouldn't be considered as offensive. (As "offensive" is too strong for it)

In fact, the expression "listen to yourself" is commonly used in the area of philosophy when there's a speak of art of communicating between humans.

philosophy reference 1

philosophy reference 2

The third

And this is a nice example of a similar usage. (Though it contains the expression and not the exact question)

in Google books