Also, regarding your question about the 'correct' synonym - the 'correct' one is the one that best suits the mood or voice of the story. Grammatically they might all work, but some will obviously fit the scene better.
Consider the following. Maybe I'm writing a story which has an evil villain. I want to write a sentence where the crazy villain laughs at the evil thing he's done.
Here's my plain vanilla sentence:
The evil villain laughed crazily. (This is okay, but not very interesting.)
Now let's play with some synonyms for 'laugh'. I'll list a few and we'll see how they work: snort, cackle, whoop, giggle, chuckle.
Plug those into our sentence and see the differences in imagery. Some will work better than others.
The evil villain laughed crazily. (boring, as we said)
The evil villain snorted crazily. (a snort is a small restrained laugh = we want a big crazy laugh for this pivotal scene)
The evil villain cackled crazily. (cackle gives a sense of madness, which goes along with 'crazy'.)
The evil villain whooped crazily. (guffaw conflicts with evil and crazy. He's insane, not a clown or a comedian)
The evil villain giggled crazily. (works with 'crazy', since we normally don't think of grown men giggling, bit it still conflicts a bit with 'evil')
The evil villain chucked crazily. (suggests mild, benevolent amusement - not what we want. Also clashes with crazy )
Of these, 'cackle' seems the best synonym for 'laugh' for the image we want to convery.
So, let's go with 'The evil villain cackled crazily.'
I could then start playing with synonyms for evil or crazily. But he's a villain, so 'evil' is redundant by definition of 'villain', and we don't really even need it. Let's just take it out entirely. A synonym for 'crazy' is insane', so I'm going to try 'insanely' in place of 'crazily'.
The villain cackled insanely.
Basically the same information as 'The crazy villain laughed evilly', but it paints a more concise, vivid image.
So, for our needs in this story, 'cackle' seems like the best synonym for 'laugh'. Change the scene, and it's no longer the best choice.
A group of young girls having fun would be more likely to 'giggle merrily' than 'cackle insanely', even though from a purely grammatical standpoint both are correct and have synonymous verbs.