Single RegionPlot with different mesh functions

One possibility is to draw the mesh in a second plot and draw it over the original one:

Show[
 RegionPlot[
  {a, regions}, {p, 0, 10}, {q, 0, 10},
  FrameLabel -> {p, q},
  MaxRecursion -> 3,
  PlotLegends -> 
   SwatchLegend[Automatic, Prepend[b, "Infeasible: " And @@ a], 
    LegendMarkerSize -> {20, 20}]
  ],
 RegionPlot[
  a, {p, 0, 10}, {q, 0, 10},
  MeshFunctions -> {#1 + #2 &, #1 &},
  Mesh -> {Range[0, 20, 1]},
  MeshStyle -> {Orange, Green}]
 ]

The code above also demonstrates how to achieve size independent mesh spacing. The only thing to watch out for is that you might need to adjust the bounds of the Range to cover the values returned by your mesh functions.

Update: It turns out we can use a simple modifications of the option setting in OP for MeshFunctions to get the desired result:

1. Boole:

RegionPlot[{a, regions}, {p, 0, 10}, {q, 0, 10},
 FrameLabel -> {p, q}, MaxRecursion -> 3, 
 MeshFunctions -> {Boole[# + #2 >= 15] (#1 + #2) &, Boole[# + #2 >= 15] #1 &}, 
 Mesh -> {30, 20}, 
 MeshStyle -> {Orange, Green}, 
 PlotLegends -> SwatchLegend[Automatic, Prepend[b, "Infeasible: " And @@ a], 
   LegendMarkerSize -> {20, 20}]]

Alternatively, specify a list of values for each mesh function using, for example, Mesh -> {Range[15, 30, .5], Range[0, 10, .25]} to get

2. Piecewise:

We can also use

MeshFunctions -> {Piecewise[{{#1 + #2, # + #2 >= 15}}] &, 
  Piecewise[{{#1, # + #2 >= 15}}] &}

to get the same result.

3. ConditionalExpression:

Finally, we can use ConditionalExpression if we use list of values for the Mesh option setting (and increase PlotPoints to remove artifacts):

RegionPlot[{a, regions}, {p, 0, 10}, {q, 0, 10},
 FrameLabel -> {p, q}, MaxRecursion -> 3, 
 MeshFunctions -> {ConditionalExpression[#1 + #2, # + #2 >= 15] &, 
   ConditionalExpression[#1, # + #2 >= 15] &}, 
 Mesh -> {Range[0, 20, .5], Range[0, 10, .5]}, 
 PlotPoints -> 100, 
 MeshStyle -> {Orange, Green}, 
 PlotLegends ->  SwatchLegend[Automatic, Prepend[b, "Infeasible: " And @@ a], 
   LegendMarkerSize -> {20, 20}]]

Original answer:

An alternative approach: Use Texture[img] as the PlotStyle setting for the desired regions using img of your choice:

colors = ColorData[97] /@ Range[4];

texture1 = Texture[ExampleData[{"TestImage", "Mandrill"}]];

legendmarker1 = Graphics[{texture1, Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
     VertexTextureCoordinates -> Automatic]}];

RegionPlot[{a, regions}, {p, 0, 10}, {q, 0, 10}, FrameLabel -> {p, q},
 MaxRecursion -> 3, PlotStyle -> Prepend[colors, texture1],
 TextureCoordinateFunction -> (Rescale[{#, #2}, {0, 10}/2] &),
 TextureCoordinateScaling -> False, 
 PlotLegends -> SwatchLegend[Prepend[colors, texture1], 
   Prepend[b, Row[{"Infeasible:  ", And @@ a}]], 
   LegendMarkers -> Prepend[ConstantArray[Automatic, Length@b], legendmarker1],
   LegendMarkerSize -> {30, 30}]]

To use a hatched image instead of "Mandrill", you can use a simplified version of the function hatchingF from this answer

ClearAll[hatchedTexture]
hatchedTexture[ps_: None, cols_: {Red, Orange}, m_: {10, 10}, 
  t_: AbsoluteThickness[5]] := 
 RegionPlot[True, {x, 0, 1}, {y, 0, 1}, 
  MeshFunctions -> {# + #2 &, # &}, Mesh -> m, MeshShading -> None, 
  MeshStyle -> (Directive[t, #] & /@ cols), Frame -> False, 
  PlotRangePadding -> 0, PlotStyle -> ps, BaseStyle -> EdgeForm[], 
  BoundaryStyle -> None]

Replace texture1 and legendmarker1 above with

texture2 = Texture[Rasterize @ hatchedTexture[]];
legendmarker2 = Graphics[{texture2, Polygon[coords = {{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
     VertexTextureCoordinates -> (coords/2)]}];

to get

Use texture2 = Texture[Rasterize @ hatchedTexture[LightBlue]] to get

Use texture2 = Texture[Rasterize @ hatchedTexture[texture1, {Black, Black}]] if you wish to be cruel and put "Mandrill" behind bars: