You wouldn't lose the uterine lining on either side, if your pregnancy hormones were in full swing, unless your progesterone level is rather low. Why not see your doctor and get your hCG level tested 48 hours apart, to be sure the pregnancy is progressing normally? They could also do a progesterone test as well to be sure your progesterone level is high enough.
You shouldn't. Menstruation is triggered by hormones that are not present in large amounts in a normal pregnancy. Those hormones are systemic, so should affect both of your uteri. That is not to say it can't happen (some women with a single uterus have bleeding/spotting during pregnancy).
Congratulations! You should probably schedule a prenatal appointment asap. Are you already taking prenatal vitamins?