The method that Amos describes is the way I go with the exception of point 6. Rather than thread the string through then wind, I wrap a couple of turns round the peg then thread the end through. That removes the guesswork in how much slack to leave, although it can be a bit more fiddly to do the threading. I do have a winder but find it a waste of time. It might be worthwhile if we were putting on a dozen windings per peg, but how difficult is a couple of turns? On another forum I've seen discussion about a winder attachment for an electric screwdriver. How many turns are these people putting on!

Advice I've read elsewhere suggests that there is nothing wrong with removing all strings at the same time and it's a good opportunity to give the fretboard a cleanup.

Not sure how long it takes to do this, what's the rush? As I said, it's the chance to give things a clean up, and have a general look over the instrument.