Are mass and weight equivalent in the same way mass and energy are equivalent?

Answer 1

no, mass and weight are not equivalant but they are related to each other according to:
w=mg

and mass and energy relationship is:
E=mc2
which means mass is one type of energy itself

Notice that g is not a constant while c is.

Answer 2

Mass and weight are proportional, in a fixed gravitational field. Varying the intensity of the gravitational field will alter the weight with no change in mass.