Yes. Although I have not personally been on a cruise, in general i think it should be as safe as other vacation options.
Yes. But make sure you consult your doctor first. There are certain age requirements inorder to travel safely with a child.
Yes. After 2 months, and after at least one round of immunizations, it's acceptable, but after 6 months and 3 rounds of vaccines would be better.
No. Many cruise lines have minimum ages to travel. A typical minimum age is 12 months but varies by cruise line. You will want to check with the cruise line before booking your travels. Although infants tolerate travel very well, the concern is how to properly care for an ill infant aboard ship. The ship's doctors are usually not prepared to handle anyone younger than 12 months, hence the restriction.
Yes. It sort of depends on the cruise and the age of the child. Children under 2 mo are not supposed to travel publicly. If they are healthy, older babies can go on cruises that have a doctor on the ship familiar with children. Most ship docs can deal with ear infections and other maladies easily. If the child is special needs, you must clear it with the cruise co. Before booking.