All guns have a minimum skill requirment. Take cowboy repeater for example. It has a skill requirment of 25. And strength requirment of 4.

If you have both the requirments and u put in more points in the guns skill, would your accuracy in VATs and outside VATs improve further? If yes, then what does the skill requirment mean?

The skill requirement needs to be met so the gun can work at its full potential. Any higher over the requirement and you will be able to use the gun above its potential. I think.

To an extent, I believe. For example, after amount of gun skill points - depending on the gun - whenever I use scope or iron-sight the gun no longer sways

If theres no improvement after meeting the requirment, im planning to make a character with 25 guns skill. Or at most 50, as that meets most guns requirments upto the hunting rifle. Going any further seems to be a waste.

The Guns skill goes beyond just meeting requirements. Yes higher guns means being able to use Miniguns, but it doesn't stop there.

With higher Guns skill the player will be more accurate, have reduced sway with the larger rifles, and deal more damage (at 100 skill level, 2x the damage compared to 0 skill

I dont think the damage part is true. There was no increase in damage for cowboy repeater from 25 to 100. And i think you get sway if you dont have the appropriate strength and skill requirment. More than often its the strength. Still i dont have any hard facts, those will be helpful.

Guns affects your accuracy- higher guns = higher accuracy. If your guns is below the minimum for a weapon, you recieve a penalty to accuracy. That's my understanding of it, at least.