I believe that France didn't open its borders to this wave of refugees/migrants but I could be mistaken since I don't follow the news this closely.
Based on what? I'm pretty uninformed here, and it seems like a lot of people are bringing up refugees. Curious why you think the opposite.
Cross border travel within the EU is like moving between states in the US, easy to miss.
I'm very positive that they check people's passports and papers in airports.
http://ec.europa.eu/eurostat/statistics-explained/index.php/...
Trains are a tiny 6%.