"Tüm gece bebek ağlayabilir"?
Why not bebegi?
It would be "his/her baby"
Why not hep gece
hep means always, so it doesn't make sense
Maybe you meant to ask why not HER gece. Her gece means every night. All night is butün gece.
And how we say the baby could cry all night
That would be 'We were able to cry.'.