Liza Jane B. answered 03/23/19
Lover of writing and teaching people how to write!
Onomatopoeia is a great tool to use in writing (especially in creative writing), but should be greatly avoided if you want to insert them directly into the dialogue. In my opinion, using onomatopoeias directly in dialogue, as your example shows, is a very amateur, fanfic genre move, and I don't mean to demean those forms of writing as they are still legitimate, but it would almost cheapen your writing if you were writing outside of those genres/forms. To show that someone is crying you should simply state it in the dialogue tags and in descriptions and actions of the character. For example, to copy yours:
"(insert dialogue)," she cried / she sobbed / she said with tears coming down. (That last one is a bit cliché, though.)
OR
"(insert dialogue)," she said. She choked out her words in between sniffles from crying.
As for your last question, there isn't really one word to describe someone calming down. You should simply describe it in your writing. For example:
"Her breathing was ragged but slow. A few tears rolled down until she eventually stopped crying. Her face was still wet and she took in a sharp breath every now and then."
Hope this helps!