I got an actual game of Pokemon Red on hand, but I'm not sure over ¤how¤ exactly I'll be able to project screen onto the computer if I only got a Gameboy Advanced to play it on. Is there any way to do it without it being terrible quality?

The best way to stream it would be to get a gamecube with the gameboy player attachment which allows you to play gameboy games on your TV. then just use a normal capture card to stream it.