Is there a single point generator G for every elliptic curve?
There often are multiple points which could be generators. In fact, for prime order curves, every point on the curve is a generator.

For practical use, a specific point is picked based on some constraints.

May curves use rules such as lowest abs(x) value or use some deterministic generator with a random seed.

I can't seem to find the login behind the G selection for secp256k1, however, but the point itself is defined in the curve specification.