Is that what you Americans call a ''pinhead''?
Ugh, sorry, I don't have the exact grit of the sandpaper in my head right now but the only thing that really matters is that you take the finest grit sandpaper available for the final step of the sanding, you don't want to have any scratches from coarse sandpaper in your stock when you are finished . What you use for the rough work to remove the finish is up to you, you can even take fine sandpaper for that task but it will take you forever.
Take your time, don't remove too much wood because wood once removed is hard to get back on the stock. Don't remove so much wood that the butt plate sticks out and don't touch the wood where the receiver sits in or where the action screws are, they need to be absolutely tight!
Don't ever sand across the grain, always along the grain or you risk a rough ugly texture and you may split and damage the surface of the wood.
In theory a completely free floated barrel is the best way to max out its accuracy, you don't want it to touch anything so nothing interferes with its vibrations. Remove wood from the barrel channel until you can slide a folded piece of paper down the barrel till it meets the receiver. That should be enough to ensure a completely free floating of the barrel.
Though I think a barrel from a 22 vibrates so little you won't notice any gain in accuracy, its more of a theoretical thing. With high powered rifles a free floated barrel is almost a necessity.
Good luck with your project, working with wood is fun and very rewarding.