It depends on your nationality.
The UK is not part of the Schengen Area, it has different visa requirements to France and their is passport control between Paris and London.
Even if you have a Schengen visa, it is not valid for the UK.
Americans can travel visa free to all European countries except Armenia, Azerbaijan, Belarus, Kazakhstan, Turkey, and Russia. A visa is issued upon arrival in Turkey and Armenia, but a visa is required prior to arrival in Azerbaijan, Belarus, Kazakhstan, and Russia.
no
All Canadian and American citizens need passports to travel to Mexico. The main reason they need the passport is to reenter Canada or the United States.
No. American citizens don't require a visa to enter Mexico. All you need is your passport.
Yes Mexican citizens do need a visa to travel to Egypt.
No.
No, US citizens do not need a visa to travel to Romania. I myself have been there three times. Romania is a part of NATO. Any country that is a part of NATO US citizens do not need visas to enter. US Citizens need a visa if they wont to travel to Russia. I was in Frankfurt and some American didn't have a visa for Russia. The Airlines flew him to Russia but when he landed the government put him on the first flight to Frankfurt and deported him.
No, they are European citizens.
Yes, you need a visa.
Yes, you still need a visa to show that you are a US resident. This is to ensure your safety.
It all depends on where you are coming from! Citizens of the EU countries can travel freely between those countries, although there are some restrictions on citizens of those countries which joined the EU most recently. Those from outside Europe may need a visa.
Yes
yes.
Yes, they do.