A homotopy of paths relative to a subspace A of X is a homotopy which fixes the points of A, so yes what you're saying is correct but unorthodox/unrigorous. And by 'graphing these out', I hope you mean graphing the homotopy at each point , because this infinite class of functions must fix A as well.
For the question on fundamental groups, the latter is true. If two loops are the same equivalence class, you can write a homotopy from one loop to another (and the converse is also true). Intuitively, you can continuously deform from one to another without passing through 'holes'. For example, since the complex plane is contractible (so has trivial fundamental group), every conceivable loop you can think of basepointed at the same point, say the origin, are in the same equivalence class. On the other hand in , the loop basepointed at 1 and the constant loop are not homotopic (in different equivalence classes) since the hole at the origin prevents the first loop from shrinking to a point.