If you are going to that region I suggest you visit a site called travelfish.org, which is by far the best website on SE Asia and it’s run by a friend of mine who lives in Bali. They have busy forums where you can ask questions and quickly get them answered by experts on every imaginable topic there. I’m happy to help more as you are planning, so let me know if you have any other questions. -Roger
Rio is probably the most beautiful city in the world as far as the setting is concerned, but it’s also famous for petty crime. As you mentioned, I think it’s quite easy to stay safe as long as you are aware of the situation. The biggest problem by far is tourists getting robbed by someone holding a knife or sometimes a gun, and they are often in groups. However, this almost always happens to people who are walking close to the water on the beaches after the crowds have gone for the day. The sidewalks along Copacabana and Ipanema are brightly lit and safe, but there is a mound of sand between the sidewalk and the water, so if you go to the water you can’t be seen by the security guards and police on the street. As long as you avoid that, and any other situation where you are alone in a dark place late at night, you’ll be fine.

People don’t think of Moscow for winter, but it’s such a wonderful time to be there. The weather is no worse than what you’d find in New York City, and the huge advantage is the people—or, really, the lack of people! No masses of tourist buses descending on the Kremlin, no huge lines for museums (of course, we ensure our travelers skip all museum lines anyway), and no crazy crowds in Red Square. Something magical happens in Moscow in the off-season: It actually becomes more approachable and less intimidating. (Plus, December falls in the heart of the theater season, so for culture vultures this is the best time to visit.)

If you’re a bit more on the timid side, you can always use Yelp and TripAdvisor to get good insights on places people have visited. As a disclaimer, some of the ratings may not be accurate on these sites, so make sure you go through and read some of the more recent reviews that people have written. We did this for a few of the places we visited and had no issues.