A dimensional check in an equation can provide a quick check about the possible correctness of the equation. For example, if you are supposed to calculate a speed, the dimensions of the result MUST be of the form [distance] / [time] (or something that you can simplify to distance/time). If it doesn't, the formula is wrong. If it does, it MIGHT be correct.
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Yes.
An ordered pair or coordinates of a point in 2-dimensional space.
It is the equation of a line in 3-dimensional space.
A line, "living" in N-dimensional space, where N is the number of variables.
There can be linear equations with 1, 2, ... variables. Each of these is different since an equation with n variables belongs to n-dimensional space.