I had a fairly bad experience using China Southern's website trying to select the seat for an existing ticket, even with Chrome, Safari, and Firefox on an OSX desktop. (Trying on a Sunday afternoon Sydney time on 7 August 2016). At least it was in English, though. Is the website optimised for a particular browser, such as Internet Explorer?
You're using the wrong website. Their global website--distinct from the English-language version of their Chinese website, which you show screenshots of--works fine without any fiddly mobile signups or Chinese error messages.
I haven't got to the bottom of how the website works, but here's some hints:
http://b2c.csair.com/B2C40/modules/bookingnew/manage/login.html?returnurl=http://b2c.csair.com/B2C40/modules/order/checkOrder.jsp, which starts off with a frustrating image of a plane above clouds, plus some Chinese text which is done as an image so I can't copy-paste-machine-translate, eventually turns into something more useful:
... which probably won't help you unless you've signed up to them.
One hassle I had with signing up was successfully getting an SMS verification code. Rather than using "Australia", and then entering in 0491 570 156 (not my real mobile number), I had to get rid of the first "0" and use "491 570 156" instead.