Uniqueness of points in Elliptic Curve addition When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!

As a general fact, a point is determined by its coordinates.

Elliptic curves are usually represented as plane projective curves, in which case they are described by a homogeneous relation between three projective variables $X, Y, Z$. If you are using a Weierstrass model then there is only one point with $Z=0$ (the "point at infinity"), so in this case yes, a point in the affine model is determined by the values of $x=X/Z$ and $y=Y/Z$. However you cannot describe the point at infinity using the coordinates $x$ and $y$.