Generally speaking Americans don't qualify for reduced admissions in Europe. Kids are free for entry at most museums. A private museum might give senior rates to AMericans.Otherwise no senior entries. (Spain is an exception) . The reduced admissions noted on sites are for unemployed (with papers and European), handicapped individuals and similar. People who can afford international travel should probably plan to pay entries to support the things that drew them to these places rather than expecting benefits supported by local taxpayers.
There are senior rates Americans can use on regional trains; they have restricted times of travel,so it is important not to travel during commuter hours or popular weekend times e.g.Sunday evening. There are not senior rates for tourists on the metro or similar local transport. There are also no Navigo Decouverte for children. Locals have official student passes. Tourists have to either buy the Visite or the adult ND.
I think it is foolish to carry the passport about when touring; I had business at the American Embassy in Paris on our last trip and was surrounded in the waiting room by frantic tourists who had missed flights or were having their trip interrupted while they replaced lost passports. The copy almost always suffices unless it is an official transaction (e.g. bank, phone purchase) Police do not stomp around demanding papers. If they were needed, they could be retrieved from the hotel safe. Just another two cents. Do what makes you feel most secure.