About wi-fi:
I've only stayed in one Bejing hotel, although a big, expensive one. That one didn't have wi-fi for sure, the only available Internet was an Internet cafeteria (without the coffee) near the reception. But things could change quickly of course. Bejing is certainly a very modern-looking city for sure (e.g. their shopping malls looked decades more modern than any I've seen anywhere). BTW you'll find lots of people speaking English in Beijing, unlike on the countryside. The problem is actually that too many want to speak English with you.. all the time. In any case shopping won't be a problem.