I too am right handed, left eye dominate. Been shooting lefty since I was 6. For shotgun, get an over/under. For rifle your bolt choices are limited to maybe two choices at your LGS or a custom but a lever action or semi auto generally works fine (just make sure to wear safety glasses). Black powder is a real PITA. The percussion cap will spatter hot stuff on his cheek - make extra sure to wear good fitting safety glasses for black powder. Archery is also a pain to find a lefty in the correct size. The good thing is that he will probably grow up to be a better shot than you - All of us lefty's are better shots than righty's
. You might consider starting him with a lever action .22 (I still have my Browning BL22 from 50 years ago) or look into a left handed Vudoo or RimX bolt 22.