There have been allegations against various hotel casinos in the DR about certain games that are scams, often run by Americans, where people have lost thousands of dollars. Yes, topless sunbathing is quite common. It's not a Dominican custom, just the fact that many of the tourists are European, and topless sunbathing is natural for them. Many hotels do not allow topless sunbathing around the pool areas. In my opinion it is perfectly safe to walk around tourist areas and towns during the day.
Most resorts have security personnel, and the beaches are also patrolled periodically.
When going into town, or out of the resort area at night, use common sense , just as you would anywhere else. Don't wear expensive jewellery or flash large sums of money around. Check with your travel agent. We always travel with a passport, personally I think it's a good idea to have one. However, Canadians are only required to have photo I.D. and birth certificate.