Originally, they were invented to provide solutions to algebraic equations, which would otherwise have no solution. Through the work of Euler, Gauss and others, the usefulness of imaginary and complex numbers in applications of periodic motion and waves was recognized. See related links.
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Imaginary numbers are only ever used when you are using the square roots of negative numbers. The square root of -1 is i. You may find imaginary numbers when you are finding roots of equations.
Yes, imaginary numbers are a subset of complex numbers.
No difference. The set of complex numbers includes the set of imaginary numbers.
No, it is imaginary. Irrational numbers are a subset of real numbers Real numbers and imaginary numbers are sets without any overlap.
Imaginary numbers are used in complex numbers that make some math simpler like electronics where there is a cycle frequency it makes the math much simpler to handle complex equations