Is it normal for my breast size to stay the same in pregnancy?

Yes. Some ladies' breasts do enlarge quite a bit- especially as delivery nears. Some increase only a modest amount in size. It depends also on where you are in your pregnancy. I would not get worried about it.
Yes. Most women will experience some changes with their breasts during pregnancy. Other women may not have a noticeable change in breast size. If you experience nipple changes or experience bloody breast discharge, please consult your physician.