You may want to ask yourself if you can get a good/excellent sound (to you) before you change the machines,nut,bridge,etc, as a nasty sounding guitar that stays in tune...Etc. Pickups are far too damn expensive these days, for my liking. :D
I had to get just one machine for my Les Paul copy, which the tech kindly gave me (free) and i cannot get the affected string to play properly, my fault as i saw it was too long compared to the old one i took in at the time but took it because i'm cheap, and so's the gtr.
Talk to the tech at your guitar shop to see if he knows what make fits LP copies well, usually a Les Paul "style" tuner is designed for Les Paul "style" guitars, so they should fit and it should be an easy job (is any job easy the first time??) , you may need a very small screwdriver for the retaining screws at the back. Usually guitar techs won't bite your head off for asking questions(especially if they think they're actually going to get money/paid :)) .Good luck to you and I hope that it turns out to be a good un .
Accuracy,you say? hmm interesting concept..