Suppose the function, y = f(x) maps elements from the domain X to the range Y. Then
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It is a function notation
With a formula, you know the variable's value, and you have to calculate the value of the function of it. With an equation, you know the function's value, and you have to calculate the value of the variable.
A linear function has and x and a y and neither one is raised to a power other than 1.
The word sought is probably "function", although in some contexts "equation" would also work.
No. It is an estimated equation that defines the best linear relationship between two variables (or their transforms). If the two variables, x and y were the coordinates of a circle, for example, any method for calculating the regression equation would fail hopelessly.