Charge density (σ) and distance between plates of capacitor given just the relationship between potential difference (V) and width of a metal slab?
Yes, the relationship of voltage and slab thickness is linear. 

The metal slab effectively brings the plates closer together. You can consider the plates separated by d to be separated by d-a when you insert a slab of thickness a. 

As long as the slab does not touch the plates, the charge remains constant.