How to fix the “Your NAND dump is probably incomplete” error on the Dolphin emulator

Yes, Dolphin supports 3D TVs. Use either the Side‑by‑Side or Top‑and‑Bottom mode.

Dolphin has 3 different 3D modes. Side-by-Side, Top-and-Bottom, and Anaglyph. The 3D settings are located in Options → Graphics Settings → Enhancements → Stereoscopy.

Side-by-Side and Top-and-Bottom 3D are what you're looking for. This splits the image in half across the screen, and should be compatible with any modern 3D TV.

As Assorted Trailmix mentioned, Dolphin also supports Anaglyph 3D, just in case you have any Red/Cyan 3D glasses lying around.

If you're interested in the history of Dolphin's 3D support, I recommend reading A Second Perspective: An In-Depth History of Stereoscopy in Dolphin. It's an interesting look into both the technical details and history of Dolphin's 3D functionality^*.

^{*Note: I'm friends with one of the Dolphin Emulator blog writers, so I may be a bit biased. :)}

You should be able to install a new CD drive yourself, so long as you find a compatible model.

Having never installed one of these on Xbox myself, I can't guarantee it'll work 100%, but here's a list on wikipedia of drives in the Xbox 360 models, if you get one of these you are probably in the clear.

There's a video here on how to replace the DVD drive, however it does note that you need to know how to solder to do it, so if you don't want to mess with it, then you might want to just get a used xbox 360 on amazon. Get the 4GB one since you can transfer the hard drive over using a little hard drive caddy you can get for pretty cheap.