Is it safe and economical to catch a taxi from Airport to hotel in China?

Yeah, generally it is safe enough to take a taxi in China. No need to worry too much about that!

For SH city, Dazhong and Qiangsheng taxi companis are suggested for you.
I just heard about Beijing's Shou Qi Group has good service.
As for Xian, the taxis are all acceptable. If you wish to know the groups, Tianzi and Qianjin groups's taxis are suggested.