I plan on getting the online adapter sometime soon, but wanted to know the answer to this before I get it. Does the adaptor have to be plugged directly in to the cable modem, or is there a way to network the connection to the system. For example, the cable modem is in the living room in my house. If the PS2 were in my bedroom, could I network the system to the cable modem in the living room, or will I have to move the system into the living room with the computer and modem?
I would recommend buying a router. You can get a 2-port wired router for around $10-$15. What you need to do is plug the router into the cable modem, and then plug the two different items that require an Internet connection into the router.
Since you're trying to connect things in different rooms though, unless you're willing to run cable all over your home (or perhaps in your walls), another option is a WIRELESS router. With this option, you'd still plug the router into the cable modem and your computer into your router, but then you'd need to get some sort wireless adapter or wireless access point to wirelessly connect your PS2 to your router. I remember seeing a wireless game adapter from D-Link for around $70 a long time ago made specifically for this purpose.
Warning: I've heard of people having headaches trying to get their PS2s working with a wireless Internet connection (that is, bigger headaches than one may usually have trying to use wireless with a PC). It might be better to have the PS2 directly connected into the cable modem and then wirelessly connecting your PC to the router. This would require moving the cable modem into your bedroom, but at least everything wouldn't have to be in the same room.
It was announced at GDC '08 that while most Nintendo Wi-Fi Connection titles will remain free of charge, some games will see a "Pay-for-Play" tag. Nintendo To Begin Charging Money For Some Online Gaming Considering Super Smash Bros.