I am left eye dominant and left handed. I have been shooting all types of right handed long guns all my life. Ejected brass, shells, etc is just not that big of an issue. My advice? Practice, practice, practice, shooting left handed, Peroid.
Shooting across the gun is never the answer, and it just looks stupid. Guns are not designed to be aimed this way. I only have one left handed firearm (a bolt action) and quite frankley I don't really see a big advantage.
So shoot the righty guns from the lefty position and you will be fine.