The shadow thing can be solved by using a sun-synchronous orbit. See for example the TRACE solar observation satellite, which used a dawn/dusk orbit to maintain a constant view of the sun.
Cooling, on the other hand? No way in hell.
Cooling, on the other hand? No way in hell.
Space is actually really cold when the sun is blocked
So, solar panels on side, GPUs on the other, maybe with a big ass radiator ...
The "dark" side of the JWST has temperature of about 40 K (-233 C)