Determine if a public key point y is negative or positive, odd or even?
Elements of finite fields don't really have a sign. But depending on context you can define a property that's different for $x$ and $-x$ (when $x$ is not $0$) and call that property sign.

Some possible choices:


A number is called a square (or Quadratic residue) if there is another number which produces it when squared. Since positive real numbers are squares, identifying square field elements with a positive sign is a natural choice.

The Legendre symbol is 0 for 0, +1 for squares and -1 for non-squares. The Legendre symbol of the product of two numbers is the product of the two Legendre symbols, which mirrors how signs behave under multiplication.

You can even define $i = \sqrt{-1}$ to get some kind of complex numbers. Imaginary values of the $y$ coordinate of an elliptic curve correspond to its twist.
You can use the least significant bit. Computing this is very cheap, you only need to look at that one bit. This approach is often used to store the "sign" of a compressed elliptic curve coordinate.
Values smaller than $p/2$ have one sign, values greater than $p/2$ the other. I believe Bitcoin uses this convention to define canonical signatures.


You can extend any of these definitions from the underlying field to the elliptic curve defined over that field by defining the sign of the point as the sign of its $y$ coordinate.