The "prog_char does not name a type", etc. errors have been fixed in this fork: https://github.com/morganchristiansson/ZG2100BasedWiFiShield/archive/master.zip
That still leaves the "conflicting declaration" errors. Why do people do that typedef crap? It always causes problems. I have to assume that at some point in the past this library actually did compile so I recommend just going back through the Arduino IDE versions until you find one that works.
When you see a library with example filenames that end in .pde that's a sign the thing is ancient. There is a pull request on the library repository from 2012 fixing the prog_char, etc. issues that has been ignored by Linksprite, pretty sad. If you can, I suggest returning that shield and getting something from a different company that actually cares about their customers. I can't believe they sell that thing for $39 + shipping. That money would buy you eight ESP8266 shields.