No, it is not. It is an arc of a circle.
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No, but any parabola can be transformed into the form y = x^2.
If you can mash the equation for the parabola into the form Y = Ax2 + Bx + C, then the parabola opens up if 'A' is positive, and down if 'A' is negative.
The answer depends on the form in which the equation of the parabola is given. For y^2 = 4ax the directrix is x = -2a.
If the coefficient of x2 is positive then the parabola is cup shaped (happy face). If the coefficient of x2 is negative then the parabola is cap shaped (gloomy face).
We can draw 3 normals to a parabola from a given point as the equation of normal in parametric form is a cubic equation.