Why don't we call them negatives or positives?
Because in finite fields there is no such thing as a negative or positive.
Does the computer chooses it randomly?
No. When you turn a private key in to a public key, you get exactly one point on the curve, which has exactly one x coordinate and one y coordinate. However, for every x coordinate, there are two possible y coordinates, so when you compress your public key down to just the x coordinate, you specific which y coordinate is the correct one.
What is p?
The field size, which for the secp256k1 curve bitcoin uses is 2
256 – 2
32 - 2
9 - 2
8 - 2
7 - 2
6 - 2
4 - 1.