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

We can also use

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

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:

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

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}]]

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

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

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}]]

We can also use

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

to get the same result.

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

Original answer:

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

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[image = 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:

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: