Are having a bad attention span and being a bit childish mean signs of adhd?

Could or not. Certainly bad attention span is a symptom of adhd, but not unique to it. Being childish or immature are usually perjorative terms for being impulsive, impatient, fidgety having low frustration tolerance, not attending to social cues. These also are not unique to adhd, but many children slapped with labels of immaturity in early grades are now recognized as adhd. Thorough assessment is needed.
Not necessarily. Those symptoms could be caused by too much sugar in the diet or from having learning or processing problems or from allergies.