1) Do I need to buy component cables or can I use the composite cables included in my slingbox solo? I don't have an HDTV but they gave me the HD DVR as a promotion.
Yes, you can use the composite cables in lieu of the component cables.
Though I would personnaly recommend both if you intend to using a mobile client. Since none of the mobile clients have stretching capabilities. You'll have to depend on the encoding resolution or viewing aspects. I've found that high-definition channels fill the screen better through the component cables, while the standard definition channels are best viewed from the composite cable.
For the Slingplayer Desktop or the Web Slingplayer, you don't have to worry since the resizing the picture is handled by your the OS's graphics sub-system.
2) Since my DVR is connected wirelessly to the Verizon router, can I hook the ethernet cable from the slingbox to the ethernet port on the back of the DVR?
No, satellite recievers haven't been designed to be used as a wireless network bridge. The ethernet port on the back of the Motorola ViP receiver was meant for cases where the WiFi signal isn't optimal.