TdF isn't raced on a nationality basis, it's raced by professional teams. Each team only represent itself, and can have riders from several different countries.
Depends on what you mean. There really aren't any countries competing, that's all done by professional teams that only represent themselves and are composed of people from all over the place. The...
They have one every year, with it lasting 23 days. The first event was in 1903, but there have been 10 cancellations due to the 2 world wars, so there have only been 98 editions in 108 years.