How would gravity affect a spring?

Yes. The spring will stretch further if the gravity increases. But remember the the acceleration due to gravity has a direction, and of course, the spring will increase its length only if the acceleration of gravity has the right direction.

You can do experiments (or thought experiments) to get a feel about it, using a bungee cord and a bag with weights, which you can switch (like small bottles of water).

To see a constant gravity force (like in real world), hang a bag with weights such that the cord is stretched a little, but not to its full potential.

To simulate a gravity force which magically becomes greater instantly, grab the bag, put an additional weight into it, and let it go. It might go up and down a few times.

Another way: pull the bag down with your hand, then let go. This way, you can simulate a gravity force which magically becomes greater, but quickly returns to normal. By doing this repeatedly, you can simulate chaotic fluctuations in gravity, or maybe if the fluctuations are regular, it can create resonance!

Yes, assuming that gravity changes slowly compared to the damping of the system spring - weight.

The effect of a sudden change of gravity would be equivalent to replace the weight by a heavier one for the same spring length. It would oscillate before rest at the new equilibrium position, with a bigger spring length.

For a constant fluctuation gravity the spring would oscillate in the same frequency. The amplitude of the oscillations would be maximum if that frequency matched the natural frequency of the system.