Is it common to say just "bratr" when referring to my brother?
I think so. We often encounter sentences like that in Czech and if it's not specified who else's brother could it be?
But then the opposite seems to be the case with Babička:
Yes, I just noticed this, too. Seems inconsistent. Maybe someone will explain...
We forgot to add 'my' in front of 'grandmother'. These things happen. Fixed now. Sincere apologies! :)