You can't; it's Ethernet or nothing.
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If nothing else is connected to the modem you can use a ethernet cable to connect the ps2 to the modem and the internet. If you also have a PC then you need to put a router between the modem and PC and also connect the PS2 to the router which will allow both the PC and PS3 to work off the same modem
No it can connect with a ethernet cable to your modem or router
Just buy any Cat 5 or Cat 6 ethernet cable. The only concern you should have is about which length you purchase. It needs to be long enough to connect from the back of your PS2 to the Modem or router
You cannot connect a Dualshock 3 to a PS2.
Your thinking about the PS3 not the PS2. The PS2 can not connect with the PSP as they don't work together