Yes. Rather than doing sit ups or crunches, you go do isometric exercises which won't impinge on the baby or hurt your back. The american college of ob/gyn has a webpage: http://www.Acog.Org/publications/patient_education/bp119.Cfm as does webmd: http://www.Webmd.Com/baby/guide/exercise-during-pregnancy which both outline safe exercises in pregnancy, as well as times when exercise should be avoided.
Yes. Almost all core-strengthening exercises are safe during pregnancy. Theoretically you don't want to be flat on your back in the 3rd trimester (better partially reclined or turned slightly to one side). Yoga is a good way. These efforts will help you throughout your pregnancy, lessening back pain, preparing you for childbirth, and toting around a baby afterwards.