Entering or leaving the Schengen Area, or between other countries there are ID (passport) checks. Usually these are done on the train, you do not have to get off the train. The Eurostar between London and Paris / Brussels is a special case. Passports are checked before getting on the train, both exit and entry checks.
A passport issued by an EU/EEA country will not be stamped on entry to another EU/EEA country. A passport issued by a non-EU/EEA country (United States, Australia, India ... ) will be stamped on entry to the Schengen Area (e.g. when you travel on the Eurostar from UK to France).
At the border control, you will be required to show all the necessary documents to the immigration officer. The immigration officer will verify the authenticity of the travel documents presented. The immigration officer will then grant the traveler entry into France by stamping their passport.
The European Union will be implementing a new digital system that will replace stamping passports upon arrival. Getting your passport stamped when traveling to Europe may become a thing of the past as a new screening system upon arrival is expected to launch next year.
On Eurostar there's no weight limit for your luggage, but you must be able to carry and lift your bags safely on your own, as there is no porter service at the station and our teams are unfortunately unable to help with luggage. Each bag can be up to 85cm long at its widest point and should be clearly labelled.
If everything is running like clockwork, however, you have absolutely nothing to lose by asking politely for an official stamp. More often than not, the officer will be happy to give you one and you'll have an extra memory to take away from your travels. It can even work in places where there isn't immigration.
And since you will be in a foreign territory, you will need a valid passport. But if you take a closed-loop cruise, the trip will start and end at a U.S. port, and you will not venture into foreign territory. So, leaving U.S. without a passport will be possible.