It doesn't need one unless you want to connect to the PSNetwork, or play multiplayer online.
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No
you need to set up your wireless router with your internet connection to your computer, and then search for an access point on the ps3 under internet setting, under set up connection, then just follow the instructions from there
no, the internet is only needed for full use of all the PS3 features and Games can be played without going online
Yes, you do not need to use a cable to connect your PS3 to the internet. You will, however, need a wireless router with an internet connection. Please note that using a wireless internet connection is usually slower that a wired one.
A PS3 slim has WiFi and wired capabilities for internet usage. It will depend on each individuals internet connection as to the slowness of the PS3.