Mesh on cross section of meshes / 2D Region on cross section of 3D Regions

One can use DiscretizeRegion to obtain a mesh:

slice = DiscretizeRegion[
  RegionIntersection[
    Ball[], 
    InfinitePlane[{0, 0, 0}, {{1, 0, 0}, {0, 1, 0}}]
  ], 
  MeshCellStyle -> {1 -> Black}
]

Or SliceDensityPlot3D to get a visual:

SliceDensityPlot3D[1, {"ZStackedPlanes", 10}, {x, y, z} ∈ Ball[], Axes -> False, Boxed -> False]

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3D surface plotters like Plot3D can plot over 2D regions. We can take slice and project to 2D:

slice2d = MeshRegion[MeshCoordinates[slice][[All, 1;;2]], MeshCells[slice, 2]];

Plot3D[x^2 + y^2, {x, y} ∈ slice2d]

If the 3D slice lies in a different cardinal plane, we'll need to project with different coordinate indices. Here's an example over a y-z plane:

reg = ImplicitRegion[(x^2 + 9/4 y^2 + z^2 - 1)^3 - x^2 z^3 - 9/80 y^2 z^3 <= 0, {x, y, z}]; 
slice = DiscretizeRegion[RegionIntersection[reg, 
  InfinitePlane[{0, 0, 0}, {{0, 1, 0}, {0, 0, 1}}]]];

slice2d = MeshRegion[MeshCoordinates[slice][[All, {2, 3}]], MeshCells[slice, 2]];

Plot3D[x^2 + y^2, {x, y} ∈ slice2d]

Note that your region has

RegionBounds[R]
{{-1, 1}, {-1, 1}, {0, 0}}

This is why the FEM mesh generation complains. See that this works:

Needs["NDSolve`FEM`"]
ToBoundaryMesh[R, {{-1, 1}, {-1, 1}, {-1, 1}}]["Wireframe"]

Note the difference in behavior for the embedding dimension of the returned object between DiscretizeRegion and the ElementMesh family of functions.

Now, let us consider a different R. You can visualize the computed values on the intersection.

R = RegionIntersection[Ball[], 
   InfinitePlane[{{1, 0, 0}, {1, 1, 1}, {0, 0, 1}}]];
Show[
 Graphics3D[{Opacity[0.5], Ball[]}],
 SliceContourPlot3D[Exp[-(x^2 + y^2 + z^2)], 
  R, {x, y, z} \[Element] Ball[]], Boxed -> False]

To refine the contours the Contour option can be used:

Show[Graphics3D[{Opacity[0.25], Ball[]}], 
 SliceContourPlot3D[Exp[-(x^2 + y^2 + z^2)], 
  R, {x, y, z} \[Element] Ball[], Contours -> 25], Boxed -> False]

You can also specify a MeshRegion

R = RegionIntersection[Ball[], 
   InfinitePlane[{{1, 0, 0}, {1, 1, 1}, {0, 0, 1}}]];
Show[Graphics3D[{Opacity[0.25], Ball[]}], 
 SliceContourPlot3D[Exp[-(x^2 + y^2 + z^2)], 
  DiscretizeRegion[R], {x, y, z} \[Element] Ball[], Contours -> 25], 
 Boxed -> False]