This stuff is ridiculously complicated. Even with the manual, it's ugly
There is a reasonable chance that your Xbox has the wrong resolution but I'm also concerned about your wiring.
There are two ways to wire the Xbox.
Looking at page 16 of your manual
1. The RGB component RCAs into the Component Video part of your Pioneer. The red and white RCAs into the Audio->DVD part of your Pioneer.
The better way to wire it.
2. The RGB component RCAs into the Component Video part of your Pioneer. The optical out of the Xbox component cable into the Assignable Optical In(TV) of your Pioneer if it is not already being used. If you are not using that optical connection for anything else, it will give you the best sound out of the Xbox but it will take some configuration in the Pioneer menu. Basically you would be assigning that optical port to the DVD (which is where the Xbox is connected via component cable). You can leave the red/white audio cable connected. It's just stereo for the Xbox but that's fine for troubleshooting.
Regardless, do not attach the yellow plug from the Xbox component cable.
Now that you have it wired, lets try and get the video working. Somehow, you need to reduce the resolution of the Xbox. I assume that you have a way to display the Xbox. Go into the Xbox system setting and reduce the resolution to 480p. Once you do this, you should be able to see stuff on your projector/receiver approach. According to page 16 of your Pioneer manual, the maximum resolution that the Pioneer can display is 1080i. So you can start increasing the resolution on the Xbox to 1080i once you have it working with the projector.
Lastly, we need to check a setting on the Pioneer. Referring to page 49 of your manual, you need to verify that Home->Video Parameter->Video Converter is set to ON. In fact I would check this first before doing anything else. It might make life a lot easier. I would also set the Resolution in the Video Parameter menu to PURE. Your epson projector can handle just about anything thrown at it.
Let me know how your troubleshooting goes.
Jotham
P.S. If you need to buy a toslink cable for the Xbox to get surround sound out of the Xbox, I would just get a cheap cable from Amazon. You don't need anything fancy.
Amazon.com: BlueRigger Digital Optical Audio Toslink Cable (3 feet): Electronics