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It's true: a curve is a curve. Did you really need me to tell you that?
Yes.
A line graph, probably.
Scatter graphs. Line graphs may be used at a later stage when there is a better idea of the general shape of the line - whether it is a straight line, a quadratic curve, a logarithmic or exponential curve etc, or one of the standard probability distributions.
Regression techniques are used to find the best relationship between two or more variables. Here, best is defined according to some statistical criteria. The regression line is the straight line or curve based on this relationship. The relationship need not be a straight line - it could be a curve. For example, the regression between many common variables in physics will follow the "inverse square law".