Where can I find a list of connections of any flight from US to any place in the world, and prices? I would like to check whether it is cheaper to book the whole trip but to get off at the connection.
I have used sites like Kayak.com you can navigate around and see where you want to go. This will scan multiple travel sights and airlines at once so you can compare.
It's unclear what you are asking. The FAA tracks about 40,000 regular flights per day (150,000 including freight, military, private planes, etc.). If we estimate that about a quarter of those international, you are looking at 10,000 flights. Each flight has 20+ fare classes so the total number of quotable prices significantly exceeds 100,000 and that's for any given day.
The sheer amount of data makes it very impractical to present all of it, so most web sites are focusing on searching after a specific target, which could be date range, destination area or range, non-stop from a specific airport (all possible destinations), etc. But it's unlikely that you'll get access to the full data set.