Good question.
I'll start with the name. They are 匙 (chí) or 勺 (sháo), literally just 'spoons'. If you really must specify (not usually necessary) you can add 汤 (tāng), meaning 'soup' to get 汤匙 or 汤勺.
In most restaurants, you will be given both chopsticks and a spoon whether you are having soup or not. They are used in pairs to eat many things. For example, if a particular food item is slippery and difficult to pick up with the chopsticks alone, then it is polite to use the chopsticks to push the item into the spoon then carry that to your bowl or mouth. Stabbing food with the point of a chopstick is a big faux pas.
No one will blink if you use it to eat your soup, but most people don't.
I don't recall ever seeing a western style soup spoon in China.
