Most complex engineering problems cannot be solved without calculus. Force related problems are a great example - how else would you calculate the force exerted on a particle a specific distance from an electrically charged wire?
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No. Calculus if a field of mathematics.
Yes, every field is an integral domain.
Calculus is mainly about limits, which in turn are used to calculate the slope of a line (known as the "derivative"; lots of applications for that), and to calculate the area under a curve (the "integral" - also lots of applications for that). For more details, read the Wikipedia article on "Calculus", or read an introductory book on calculus. As prerequisites, you should be well-acquainted with high-school algebra.
Speaking as someone who is in the engineering field i can tell you that calculus serves no purpose at all for me. Everything is just done with a calculator or a computer so it's all just a bunch of hogwash they put you through for Engineering school that you will never use in the real world unless you are to become a program writer or work in physics.
Leibniz and Newton are usually considered the most famous in the field of calculus.