are all on the plane x+y+z = 0 and are located inside different small cubes.

Clearly this plane cannot pass through the cubes defined by the entirely positive or entirely negative octants, although it does touch their common corner at (0, 0, 0).

I can rotate the plane up or down slightly so that it now passes into one of these two octants but remains inside the 6 cubes detailed above (which will be the case if the tilt is small enough). This plane therefore can intersect 7 of the cubes. This slices each one into two pieces giving a total of 15 pieces of cheese.

What about a more general plane?

With ax+by+cz=0 the choice of signs of a, b and c means that there will always be (at least) two cubes missed out by this slice.

Changing the right had side to a non zero value ax+by+cz = d will not affect the orientation of the plane and will move the plane into one of the missed out cubes and away from the other missed out cube (to see why note that one of missed out cubes is above and one of the missed out cubes is below the plane). Therefore we can never construct a plane which intersects all 8 smaller cubes and thus the maximum possible number pieces after 4 slices is indeed 15.

For the second part we can make some deductions, using the following ideas