A router is not part of the PS3 hardware, you must purchase a router if you need one.
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You can forward ports on your router for your PS3's IP, add the PS3 to the router's DMZ, or (although just this isn't the best fix in my experience) turn UPnP on on your router
You can use a wireless connection if you have a wireless router
Yes
The button is on your router
Router problem. Reset both your router and your ps3 then try to connect again. Make sure you have a good signal.