How to get rid of "fringes" in 3D plot?

The fringes stem from the fact that the disk is discretized into a triangle mesh and its boundary edges are not very short (and the fact that $\sqrt{1 - r^2}$ is not differentiable at the point $r =1$). You can avoid this, e.g., by using a different parameterization:

s1 = ParametricPlot3D[{0, a, 0} + {r Cos[t], Sqrt[1 - r^2], r Sin[t]},
  {r, 0, 1}, {t, -Pi, Pi},
  Mesh -> 21, Boxed -> False, Axes -> None, ColorFunction -> myGray, 
  PlotPoints -> 100
  ]

You can restrict x and y to Disk[] using RegionFunction:

s1 = ParametricPlot3D[{0, a, 0} + {x, Sqrt[1 - x^2 - y^2], y}, 
  {x, -Pi, Pi}, {y, -Pi, Pi}, 
  RegionFunction -> (#3^2 + #4^2 <= 1 &), Mesh -> 21, 
  Boxed -> False, 
  Axes -> None, 
  ColorFunction -> myGray, 
  PlotPoints -> 100]

Doing the same for s2 thru s6 we get

ImplicitRegion[] works better than Disk[] (but why?):

ParametricPlot3D[{2.3, 0, 0} + {Sqrt[1 - x^2 - y^2], x, y},
 {x, y} ∈ ImplicitRegion[x^2 + y^2 <= 1, {x, y}], Mesh -> 21, 
 Boxed -> False, Axes -> None, ColorFunction -> (GrayLevel[1] &), 
 PlotPoints -> 100]

Update: Another approach is to control the discretization of the Disk[], the boundary being the most important element in this problem:

disk = BoundaryDiscretizeRegion[Disk[{0, 0}, 1], MaxCellMeasure -> "Length" -> 0.001];
disk = DiscretizeRegion[disk];

ParametricPlot3D[{0, 2.3, 0} + {x, Sqrt[1 - x^2 - y^2], y},
 {x, y} ∈ disk, Mesh -> 21, Boxed -> False, Axes -> None, 
 ColorFunction -> (White &), PlotRange -> All]