If the robot is connected to a network, it does not matter if you are controlling the bot via a hard wired or wireless device.
You will need to look up the Dashboard Interface, which can be used to send basic commands (play, stop, pause, etc) and script commands (any bit of code that the robot can execute, like movements etc). It's completely possible in Python as well as pretty much any other language, as the protocol is just a plain vanilla text TCP socket.
That's correct.
The robots cannot connect to wifi directly... but you could plug it in to a wireless router or something, sure.
Hello
if it is for industrial use, you should use a product certified by universal robots. ARCS BASIC Airgate
If it's for industrial use, you should use a wired connection. lol 
r00t8 we built a cloud based app that allows you to control and monitor UR over the web. It's called Tend and you can find more information on the website: https://tend.ai.
You can see a demo at the Manufacturing & Technology conference next week in Cleveland. Tend will be in the UR booth showing it off.
<disclaimer> I am one of the co-founders of Tend.
