mrs. emily - several years ago i planted culinary ginger root that i bought at the grocery too. i'm in zone 8, n.c., and i planted mine directly into the garden bed and it has proved quite hardy, dieing back to the ground at first frost, it resumes growing when the soil temp gets back up to around 60 degrees in the spring - in a pot, you can probably keep it evergreen in a greenhouse or a sunny window.
as to division, you can, if you so desire, divide the ginger between the 2 shoots. this might best be done after the plant is well established, if it is not already. i have grown gingers in pots as well and they do very nicely that way too - they actually don't seem to mind being fairly pot bound.
so far as blooms go, culinary ginger is the only ginger i have ever grown which never has produced a bloom -
beautiful foliage, but no flowers, dag-nab-it!!!