I actually don't provide good support for this product. This is sold as a diy kit and I make it as clear as I possibly can on the page that I don't provide support for it. Most of that is because I can't really support other people's soldering skills, so I recommend if people aren't moderately experienced they should get a teensyboy which is fully assembled and tested.
One major issue here is you're not using the right code. Get the right code from the product page, that will eliminate the constant mode switching.
The Gameboy color will provide power through the link port so you don't need to power it through USB (in fact you shouldn't). It's not unheard of for the Gameboy to brown out when you initially plug the arduinoboy in, so I'd have it connected before you turn the Gameboy on.
I'd guess fixing all the stuff I mentioned will fix this. Specifically:
Flash the correct code (link on the product page)
Don't plug in the USB when it's connected to the Gameboy
If that's doesn't work then it's probably bad soldering, so double check all your joints for cold joints or bridging.