I was 1 cm at 38 weeks, walking daily, and working till the day before and went into labor 2 days before my due date. It's impossible to tell how soon some women are at 1 cm for a month but it's promising you'll go on your own and hopefully not way past due date.
Not necessarily. I was as active as my body would allow, was having sex and bouncing on an exercise ball for hours and 2 days past my due date my cervix was not thinned or open. I ended up getting induced, but it could happen for you any time. Keeping active will help. Your baby may not come on the due date. My doctor said only 5% of babies ate born on their due date.