As far as I am aware... Remington and Winchester cases are not annealed from the factory.  Privi cases are, however and if you are using .303 British cases, then you should be good to go from the start.

That said, I'm not by any means the expert here as far as metallurgy goes, so I can only offer lay advice... but I believe that annealing the cases once should be sufficient.  Annealing the case necks makes them pliable and more easily repeatedly resized without stress cracking (Brass, although soft in nature, can become brittle with repeated working AND heating, ala firing).

I know there are several accepted methods for annealing the case necks, and I won't dispute one method over another... but the method I use is this:  I chuck my case up in my drill using my LEE case trimmer adapter.  I then spin the case while applying a blue-tipped flame from my butane torch until the brass changes color down to about 1/4 inch past the shoulder.  I then loosen the adapter and let the case fall into a metal bowl to cool naturally.  Some quench the brass but my feelings are quenching can cause brittleness (you quench other metals to harden them!).  Using the drill allows me to heat the brass evenly, rather than obtaining hot spots.  This method seems to work well for me, and I've got almost 20 reloads on some of my Krag brass!

As for expanding the case necks to .358... I've never done that, so I really can't comment on it.  I defer to others with more experience in that department!