Learn English – Meaning of “doodle let me go” and “yaller girls”

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.

It's a somewhat idiomatic usage. Literally, [You can] trust me to [do something stupid, is effectively a rueful admission that it's typical of me to do stupid things (so it's no surprise that I just did it again).

It's not only used when the speaker is calling attention to his own (typical) action, so...

"Trust you to do something stupid!"

...is an idiomatic colloquial usage meaning You've just done something stupid, which is typical of you.

Also note that the "typical" action needn't always be an obviously bad thing. I can't easily track down a written instance, but...

"Trust John to throw a great party!"

...could reasonably be said admiringly by a guest if John's party is obviously going really well, and everyone (or at least, the speaker) knows perfectly well that John's parties are always great fun).

I should perhaps point out that in every context I can imagine, the object of the verb (you, John) is always heavily stressed in speech.