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Yes. (The answer may be complicated somewhat in non-Euclidean geometries, but it's possible in Euclidean geometry if the lines are parallel).
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In the mind. A plane is an abstract concept: it is perfectly even (flat) and infinitely long in every direction. A flat surface, such a s desktop or sheet of paper may model it, but these will only be approximations.
Yes, in geometry, the concept of space includes three dimensions: length, width, and height. Soils, as physical entities found on Earth, also exist in three dimensions. This means that soils have depth, width, and length, making them three-dimensional objects.