Should 3/8 and 5 and 1/4 inch wide manufactured hardwood be
installed with a stapler or should it be glued down? I have heard
both ways as the width seems to be the issue. Nail down is much
faster and has very little mess to be cleaned up, but some say that
if nailed down it might not hold as well as the glue down. Thanks.

With a floor that thin and wide, gluing will offer superior results. With stapling/nailing only, you're sure to get some "snap, crackle and pop".