China doesn't have agreements with BT, AT&T etc which allow it to tap fibre in our countries at will. I'm sure they try some tapping, but they can't do it on the scale that GCHQ and the NSA have been outside China.
Sure, but they don't have the "home field advantage" that the NSA does, whereby much of the core internet infrastructure is housed in the US. I forget the exact number, but something like 70% of the world's internet traffic transits the US. (they mention this constantly in NSA-related articles)
Are they? That's a very complicated thing to pull off and China isn't known for having the most advanced Navy (e.g. they can't secure their own oil tankers in the Persian Gulf) and they only have a small number of submarines.
This is exactly why Australia is very leery of letting the Chinese telecom hardware manufacturer Huawei have any of the contracts for networking hardware on the nascent National Broadband Network -- they are suspected of having ties to the Chinese government / army: https://en.wikipedia.org/wiki/Huawei#Security_concerns
My sister in law works for Huawei in Kenya. Her job (so far) has largely involved ripping out Siemens made mobile-telephony infrastructure and replacing it with Huawei-made mobile-telephony infrastructure. Such are today's instruments of empire.