Should a multi-language website have different privacy policy and terms of use content for the languages it implements?
No, absolutely never. Even if you opt to have translated versions of your terms of use available, every single version of your terms needs to include all of the same content. You cannot guarantee a French user will read the French language version. If there are any country-specific exceptions or changes to your terms that need to be integrated, it needs to be included everywhere.
When a website owned and served in Europe is decided to support Chinese language, is it necessary to hire a Chinese lawyer and re-write policy and the terms on the grounds of Chinese laws or just handshaking with a translator is enough?
As a general rule, you should. There are plenty of laws out there that apply to anyone doing business with you in their country regardless of where you are located. Not only would an attorney in another country be able to tell you if there are other things that need to be mentioned in your terms that are specific to that country, but they're also the most reliable source for translating the terms if you opt to go that route. A specialized attorney will probably have a trained paralegal or clerk that is familiar with the intricate details of translating legal documents - it is not something you should entrust to an ordinary translator.
Translating terms can be very tricky. Any subtle mismatches between each language can cause immense headaches for you down the road. As mentioned earlier, you cannot guarantee which language a person will be reading the terms in, and tiny mistakes can give users the option of doing something you really don't want them to do. They can easily just point at the terms in such-and-such language and you're just out of luck.
Some companies go so far as providing subtext to clarify the jurisdiction and which language is the "official" terms of use document. All others are provided simply for convenience and are not binding. But even if you go that route, you should still consider hiring a professional for the translation so that you are not confusing users who cannot read your primary language and otherwise would not learn about the real terms of use. Providing a translated version "for their convenience" that is too different from the binding terms could just as easily get you into trouble.
It's akin to posting an English sign that reads "No Smoking Allowed" and a French sign that translates as "Smoking Discouraged" - those don't mean the same things and you can't blame a user for interpreting the French sign you provided that was supposed to say the same thing.
