A false negative would be much more devastating...to tell you it is nothing when it is something. Odds are quite high in your favor for it to be nothing. Lumpy breasts are common in women your age, and in general, 80% of lumps are benign. Only 5% of BC is inherited, so "running in your family" may or may not have an impact. How old were the women who had it and what relation are they to you?
As for a "false positive"...I would not worry. In my case, at age 30, the ultrasound was definitive, whereas the mammogram showed nothing. If the u/s looks like anything, they will biopsy it to be sure. In fact, you might ask about biopsy anyway...while the u/s is good, nothing is sure till you examine the cells.
Good for you for seeing your doctor! And don't let them turn you away just because you are "too young".