There is a ton of info over at the following site on the 338's if you feel up to doing a little searching and reading. Long Range Hunting Online Magazine
Some find that RL-22 and 25 did well, some find H-1000 to be better. If it were me I would lean hard on the H-1000 and be done with it. Plenty of others use Retumbo and a couple of other powders, but like mentioned above, nothing etched in concrete here. If you read through enough post though you will probably see more references to Retumbo and H-1000 though. Either way stick with the manuals and work up from there.
Of note, you might also read through some of the topics regarding the wildcat version 338 Edge while your there. It was developed by Shawn Carlock and is very close in ballistics to the Lapua. That said though it uses the 300 RUM case. I'm not suggesting you try to duplicate the load data, but if your interested in re-barreling at some point this is an excellent one to move to from the 338 RUM. You would however need to acquire some 300 cases however, as the 338's are a bit short for the Edge.
As for a stock, well while your perusing the above link there are stocks aplenty in those threads. The real bottom line is how much do you want to put into one and what are your intended uses.
Hope this helps.