First of all, I wish you good luck trying to get a Chip into a GBA housing, unless it's the GBA SP. But personally, I think the SP is a bad choice because it's not very comfortable. If you want to try fitting the Chip in a GBA housing, first thing you should do is remove the headers, the USB port, the headphone/video connector and the battery connector. Maybe even the micro-USB connector. And of course the plastic back What remains could fit in the cartridge slot with a bit of luck, or in the battery bay (maybe you could find a LiPo that fits in the cartridge slot). You have to design PCBs for the buttons because it's nearly impossible to reuse the original GBA PCB for that. All connections to where once connectors used to be now have to be done with thin wires soldered straight to the Chip (tip: copper-enamel or kynar wires).
So be prepared for some soldering and PCB designing jobs!
I think it doesn't matter when a display doesn't support RGB666. If the display theoretically works, but expects RGB888, you just don't use R0/1, G0/1 and B0/1 and shift the other bits (ie. R2 becomes R0, R8 becomes R6).
I'm not a Linux guru, but from what people are doing with eg. a Raspberry Pi, I think that rotating the screen 90 degrees should not be a real problem and can be set so the Chip will already boot correctly.
For soldering the .4 pitch connector, you need a hot air soldering station. I strongly advise you to use the Adafruit breakout board instead.
Considering the costs of the parts and the total costs of the project, it might well be that the Chip turns out to be the cheapest component. And the completed Chipboy will probably cost 4-5 times as much as a good used GBA will cost. But hey, this is hobby!