Is the second hand independent?

This is such a basic question, and I feel like it must have an obvious answer—but I’d never thought about it before and I’m not finding a convincing explanation online.

If you have a hacking movement and can (as most of us would do) stop it when the second hand gets to 12, then set the hour and minute hands, does that mean that the second hand runs independently from the other two?

In case my meaning isn’t clear: when you turn the crown, the minute hand moves and the hour hand does too, following the movement of the minute hand, so when the minute hand is at :30, the hour hand is halfway between two hour markers. The hands are obviously geared in a way that links them precisely. But that cannot be true with the second hand, right, because the second hand could, theoretically, be pointed at 12 and then I could set the minute hand to be anywhere I want it to be, even between two minute-markers.

Does this mean that the second hand is always “doing its own thing”? I get that its accuracy is still connected to the same drivetrain as the other hands, so it’s not like one could have a watch where the seconds run fast but the minutes and hours are accurate or slow or something.

If I’m right about this… is the same also true of non hacking movements? And what’s the logic behind this unlinking?

The hour/minute hands in setting mode are normally moved via overcoming the friction between two parts (often the hour wheel and the canon pinion), so in effect you are overriding the normal drive train. With a hacking movement, it stops the balance rotating when you hack, which means in effect you have stopped the watch. You overcome the friction in the drive train to put the HM hands in the right position then you start the watch again.
In non-hacking movement it's exactly the same operation, except that there is not a mechanism to stop the balance.

Very helpful—thank you.