No. Most women feel more comfortable on their side, however, sleeping on the back is not problematic. If you feel short of breath while on your back, you should turn over. Occasionally as the uterus starts to enlarge it compresses the vessels which return blood to your heart, which can make women feel short of breath.
Shouldn't be. You may find that is the most comfortable position for you . If you sleep well it is good for your baby. Some women find sleeping on their side with a pillow betwen their knees is also comfortable. The baby's activity level and position will really affect how / when you are comfortable to sleep. Good luck.