Would dementia show up on MRI or cat scan?

Not necessarily. Mri and cat scans show anatomical brain changes only, not functional ones. If a dementia is far enough advanced to change brain structures, it would "show up" on these scans. However, there may be a long decline in brain function before that point is reached. There are functional scans which might detect changes before anatomic anomalies can be seen. A normal MRI does not "rule out" dementia.
Yes. Generally brain imaging will show atrophy/shrinking of the brain. Imaging may also help to rule in or out other causes.