There are so many that it is impossible to name them all here. Essentially, anything that will have two solutions, will apply. The beauty is that x could stand for anything. For example, it could be used to calculate a series of possible angles that satisfy a trigonometric function, which could be describing, say, the behaviour of a rounded surface. This in turn could be applied in engineering, design, computer graphics. When using imaginary numbers, uses include applications in electronics and many other things. X could also stand for another entity which is not as simple as x and which in turn could represent another even more complicated entity. (ie x=y^4+c; y=t-s; etc etc).
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All quadratic functions with real coefficients can be graphed on a standard x-y graph. Not all quadratic functions have real roots, maybe that's what you were thinking of?
There are linear functions and there are quadratic functions but I am not aware of a linear quadratic function. It probably comes from the people who worked on the circular square.
There are different methods of using quadratic functions depending on the equation.
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Yes