Yes. Γέννηση is the birth, whereas γέννα is the delivery, the labor, the birth. So you can say "Η γέννησή του παιδιού ήταν δώρο"=The child's birth was a gift. "Η γέννα ήταν δύσκολη=the delivery/birth was difficult", also "the birth went well and the mother is fine=η γέννα πήγε καλά και η μάνα είναι μια χαρά"
Or to put it differently: γέννηση is "birth" from the point of view of the child, and γέννα is "birth" from the point of view of the mother.
Because that's a translation for "the birth, the birth" -- but there's only one instance of "the birth" here.
You can answer with η γέννα or with η γέννηση (both are accepted) -- but not both at once. Pick one :)
"birthday" in Greek is τα γενέθλια -- a plural term like the name of some other feasts such as the Latin "Bacchanalia, Saturnalia".
I believe it's related, but not very closely, as this has the γεν- stem with one nu rather than γενν- with two.