GameBoy Advance SP won’t load anymore

No, it is not possible to be able to link Game Boy Advance games on the Nintendo DS. Unless the games you want to play have the multiplayer feature that only uses one unit, you can only play them as one player on the DS, as the GBA games are not able to use the DS wireless features and the DS does not have a connector for a link cable.

When the Xbox 360 starts up, there's a little ring of lights around the power button - are these lights all off, some green, or are any of them red?

If there are green lights, I'd check the connection between the Xbox 360 and the TV, for a start. The TV probably needs to be on a specific input, depending on how the two devices are connected. You might also try a different cable or a different input on your TV, if you have multiple inputs that could support the Xbox. Some Xbox 360s support HDMI, which is probably the best way to go. There's also a cable that generally comes with them that can be used for either "component" or "composite" inputs on your TV.

If the lights aren't coming on, it might be a power issue, so check the connection to the power supply. However, you mention it sounds like there is activity, so this seems unlikely.

If the lights are red, there's not much that can be done. Unfortunately, there's not really any "user serviceable" parts inside an Xbox 360. It's likely that the box is out of warranty, so Microsoft would probably want to charge you more than it's worth to fix it, but contacting their support might be worth a shot anyhow.