The answer is yes to both of your questions.
What you need to know is that it is normal for the uterus to contract throughout pregnancy and some women can feel it early on.
Talk to your care provider about the differences in Braxton-Hicks contractions and true labor. You can to a quick search online, too. Try to allow yourself to stay calm, there's nothing dangerous or negative about this...Braxton-Hicks contractions are very helpful, they're like practice or dry runs for your uterus.
One last bit of advice from every single fiber of my being...whether or not you're planning medication in your birth, contact a doula (several and interview them)!! It's the best gift you can give yourself, your baby, AND your husband!! Do some research on that, too. Look up Doula studies.