1) Yes, very capable out to and past 500yds.
2) For your usage, you won't need perfection when it comes to 'counting clicks'. Just about any reasonable scope will perform well enough. I'm sure folks with Nikon experience will chime in.
3) I've seen a LOT of rifles that shoot like a house a fire with their original stock, sometimes with just $25 worth of materials and an evening of elbow grease to make sure it was bedded correctly. If the stock is comfortable and the gun shoots, there is no reason to change it, unless you "want" to.
Edited:: If you don't NEED a different stock, take the money and buy a better scope. Remember, rule 1) Buy the best scope you can afford. Rule 2) use the rest of you money for a rifle.
4) No. But don't forget to wear hearing protection "EVERY TIME" if you hunt with a muzzle Brake
. For the 338 Win Mag, I wouldn't add a brake if it didn't have one already.