Yes, a good cry is somewhat idiomatic. The phrase has been around for quite some time, too. An 1869 magazine article read:
Completely overcome by chagrin and disappointment, I had a good cry over it.
and in an 1887 issue of Harper's:
How often in her own room, distracted between those conflicting fears and hopes, she sat down with her face between her hands and had a good cry over it!
More recently, in a 2004 novel by Steve Tesich:
He had a good cry over it and it made him feel better. Made him feel good again. Good and heartbroken. Good and scared. But basically good. He was feeling good about something again.
So, what is a good cry? The website AgingCare.com offers a hint:
Researchers have proven what many caregivers have already figured out on their own: sometimes there's nothing like a good cry to make you feel better. Crying is cathartic.
We can cry about a lot of things, but a "good cry" will make us feel somewhat rejuvenated and emotionally cleansed. When comparing the improved moods of criers as compared to their more "stoic" peers, one researcher said:
This pattern is often found in retrospective studies where people are asked to rate their mood levels after having experienced a good cry.
Going back to your original sentence, it's hard to say for sure why the person cried – not without knowing what was in the letter. Whatever emotion triggered the tears, though, we know it was a good cry, meaning that the cry was deep, and that some emotions were released to the point where the person felt "cleansed" afterward.
No, they are different!
take a look is an idiom which means "to observe or examine someone or something."
On the other hand, looking at someone is merely an act of seeing someone. The purpose of the former one is different.
You look at some product as a normal physiological gesture but if you 'take a look' at the same product, you have a purpose to study/observe it.
Best Answer
This may be a little bit of a stretch, but I see it as a combination of two expressions: take a good look at something/someone and turn your back on something/someone.
We say "take a good look at [something/someone]" when we want the person to remember the thing or person being looked at. This is often used when the thing or person will be missing, removed, denied, or some similar state. For example, imagine a kidnapper taking a parent's children. The kidnapper might say
Figuratively, when we turn our backs on someone, we reject them:
So, although X will literally see Charlie's back when Charlie skates away, I think it can also be interpreted figuratively. I would say it means something like remember me because I will reject you, or more plainly, remember our friendship because we are longer friends (since X stole Charlie's skateboard, according to Charlie).
I'm just guessing that the relationship is "friends", but it could be something else, like teammates, colleagues, or whatever the case may be in the text.