Hola: We just booked the Punta Cana Princess for our 9th trip to the DR. As to your questions:
1.) If you eat and drink in moderation you should be OK. Take precautions: bring imodium, pepto, and maybe an antibiotic to be on the safe side to cover any and all cases. Sometimes I start taking a Pepto (or generic) pill a couple of days before my trip and during the trip. In all of our trips I have only been sick once and it was my fault. I ate a hamburger that wasn't fully cooked.
2.) Don't drink the tap water in your room. They give you bottled water in your room fridge. Use it to brush teeth!! The water in the restaurants and food prep water is all purified.
3.) I wouldn't go out alone at night, but during the day you can be a "tourista" walking around the nearby Bavaro area and anywhere up and down the miles long beach.
4.)It depends where you are flying through. We try and fly thru San Juan which is a small airport, and customs is a breeze. Miami is more of a hassle, but overall no problem as long as you follow the rules about what you can carry on the plane, and have your passport, etc.
5.) Bugs. This is the tropics and the tropics do have bugs. But that said, all the resorts spray, and you see the occasional bug here and there. Some people complain about the mosquitos, but in all of my trips I have not been bitten. And up here in Maine I am usually their main course.
6.)Being a "first timer", going on one of the truck safasi's is a good choice. You are on the back of a truck and you go thru the countryside maybe seeing a sugar cane processing, and you usually stop at a "locals" house to see how the Dominican people live. It gives you a good overall fel for the country. The other great trip is the Saona Island trip. But that is a long day, but everyone loves it. You take a bus from your resort and go to bayahibe (1 1/2 hours) hop on a speedboat and/or catamaran and go to Saona with a stop along the way. Spend a few hours on Saona Island and then take a boat back. You go one way by speedboat and the other way by catamaran.
Hope this helps.
Mike & CYndie - The Maineacs