What percent of the world's airline operations are in the USA?
It may be a bit hard to find this data for many different countries, especially if you want to separate takeoffs and landings.

According to this data for the US, there were 757.1 thousand flights in May 2014, including both domestic and international. Of those, 683.0 thousand were domestic, and 74.1 thousand were international.

Lots of information is available here. This indicates 8.3 million domestic and 1.4 million international flights in the US during 2013. This includes all carriers an all airports, including domestic all-cargo flights.

The best data for Europe I could find is only in passenger count (the more common statistic). It looks like you have to request access to get more detailed information.

However, this collection of data from Eurocontrol showed 9.548 million "IFR Flight Movements" in 2012. This would include all IFR flights, regardless of their type. A graph at the bottom shows figures for 2008. This shows a little over 8 million internal, and a little under 2 million arriving/departing. Since a domestic flight would be two movements (departure and arrival), this is 4 million and 2 million flights. As expected, Europe has a much higher percentage of international flights (about 1:2 international in Europe, versus about 1:9 in the US).

For domestic and international traffic in Australia, there are 702321 domestic flights and 162207 international flights in 2013, for a total of 864528 flights.

Data from the UK show that there are about 800000 domestic and 1000000 international flights per year.

You could also order a year of World Air Transport Statistics for only $799.