Tracing – How can I "chop" a plane from a VectorPlot3D to get a flow path?

Well, for starters, the $ x $ and $ y $ the solutions are independent of $ z $. In addition, I understand that there is a symmetry, but it is also useful to draw the negative values ​​of the variables (because $ (x> 0, y <0) textrm {or} (x<0,y>0) $ has a different behavior of both negative or positive:

StreamPlot[{-x - y, 2 y}, {x, -1.2, 1.2}, {y, -1.2, 1.2}, Axes -> False, 
  FrameLabel -> {Style["x", Bold, FontSize -> 24]Style["y", Bold, FontSize -> 24]}, StreamColorFunction -> "Rainbow",
StreamScale -> {0.1, .7, None}]

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Here is a view looking at the $ xz $-plan, with your code and very minor adjustments,

    VectorPlot3D[{-x - y, 2 y, z}, {x, -1.2, 1.2}, {y, -1.2, 1.2}, {z, -1.2, 1.2}, 
   AxesLabel -> {Style["x", Bold, FontSize -> 24]Style["y", Bold, FontSize -> 24],
Style["z", Bold, FontSize -> 24]}
VectorColorFunction -> "Rainbow", VectorPoints -> 5,
VectorScale -> {0.1, .7, None}, ViewPoint -> Before]

The main "addition" is Point of view-> Before, which I have just looked at in the documentation.

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