Glad you got it diagnosed, and hopefully you can get your data. Always back up! :) To clarify, mister790 is correct, it means the system cannot find a volume to boot from. Even if there is no startup disk set in preferences, the system will still look for something else to boot from and then go with it. For example, if you set the startup disk to the OS DVD and restart, it will happily boot to the DVD. Now restart again and hold down the mouse button to eject the disk on reboot. It will eject the DVD before trying to find a volume to boot from, meaning that the startup disk selected (the DVD) will not be valid. It might flash a question mark at you for a second, but very very soon after will find the OS on the hard drive and boot from it automatically. When I did it on my laptop, I never even saw the question mark. It just booted to the HD and was happy. Insert the DVD for the next boot and it will boot from that, as it is still the selected boot volume.
Actually, you need CAT6 for gigabit ethernet. CAT5 and CAT5e are for 10Base100 Ethernet. Also, make sure that whatever is at the other end is also gigabit. If you have a device at the other end that is only 10/100 it will show up as being 10/100 instead of gigabit.
rdklincorporated is 100% spot on. OS X still can get bogged down with things starting up and it still needs to be defragged occasionally. Check the Login Items tab on your User account to see how many things automatically start up when you log in. See how many you can eliminate. Make sure that at least 10% of the capacity of your hard drive is free space (10G on a 100G drive, for example). The OS needs this to use as swap space and not having that slows it down tremendously. As doing backups should now be done by everyone, doing a complete erase and install will not hurt a thing and can rejuvenate an older machine.
Generally speaking you do not lose much or anything from your iPod if you have to replace a logic board. The main reason for this is that generally everything is backed up on your computer, and after replacing parts on the iPod, you restore and everything is back as it was. If you got something directly on the iPod and had not synced it with the computer, then that might be lost. Other than that you are at low risk if you have synced it. I only added this to more directly answer the title question.
It certainly sounds like one. There is a possibility that it's the hard drive, RAM, or even the optical drive causing issues with it. With the video symptoms, though, I think the logic board is indeed the problem.
If you have a router, or one of the newer DSL modems - do NOT try connecting using PPPoE directly from the computer. You will then be logging in when you are already logged in. The router or DSL modem should handle the PPPoE connection, not your computer. As to why you might not be connecting to the internet from the Linux side, try entering in terminal "ifconfig" and make sure you are getting an IP address. You might even try setting a static IP address and see if that helps the situation. Some (older) home routers can't deal with a computer restarting into another OS. It sees the mac address as the same, but for some reason doesn't like the different OS. Restarting the router seems to help on these. I have seen it mainly on old Linksys routers.
I have found that Onyx is an excellent app for keeping my Macs in top shape. It's also free, which helps. There are many little tweaks in there that will get your system running as well as it can. Hope it helps.