Is it acceptable to request a brain scan out of curiosity?

No. Unless you want to pay out of pocket and find a physician that would humor your interest (I don't think it is appropriate however in that it is an inappropriate usage of healthcare resources).
Not usually. There are two main types of "brain scans", CT (or cat scan) and MRI. Both tests are expensive, and it is unlikely that an insurance company would pay for them, just out of curiosity. There is some risk of radiation exposure with a CT scan of the head (although small, 0.01% risk of later developing cancer because of the scan).