Yes, which is why one needs to be prepared to show their visa again. However, I have not had this happen to me. Also, most of my flights are direct. I rarely choose an airline involving a layover and if I do, I'll spend 24 hours there as I hate long flights.
The embassy asked for it? That is strange. Still, the airline shouldn't. On a tourist visa, yes, I can see that. All other visa types should not require one.
They will if you have a visa. It's very simple. As a passenger, you are responsible for your visa business. You must be prepared to show your visa to airline staff upon check-in without them asking you. Look at the staff and if they miss your visa, show them.
Nothing wrong with being asked how long you will be staying for, particularly for visa-exempt folk.
US immigration asks this question (or similar ones) of most arrivals and it's the same in the case of EU/Schengen, UK, Australian and New Zealand immigration.