On busy travel days, such as near Christmas or before Thanksgiving, do airlines add more flights or is the number consistent with any other day?
Yes, airlines adjust capacity to meet holiday demand. For example, Michael Goldstein reports in Forbes that for Christmas 2017 travel season, defined by Airlines for America (the industry consortium of major U.S. airlines) as December 15 through January 4,
To allow the extra 80,000 people a day to fly, the airlines will offer nearly two million additional seats, (91,000 per day), by adding flights and utilizing larger planes.
If you have traveled on a particular route during and outside of busy travel seasons in any part of the world— Golden Week, Semana Santa, Chinese New Year, Carnival, etc. — this may be obvious. For example, in the domestic U.S., there is extremely low demand for flights on the afternoon of Thanksgiving Day, and many flights that would normally operate on a Thursday are not scheduled. On the other hand, the Wednesday and Sunday surrounding Thanksgiving are usually the top two travel days for the whole year, so extra flights are added.
(I've personally experienced the aircraft swap as well. I specifically booked a flight ordinarily operated by a 3-class international configuration 767 for the Sunday after Thanksgiving only to find a 2-class domestic configuration 777 waiting at the gate.)
Anyway, these adjustments are not limited to holidays. The major airlines shuffle aircraft and crews around constantly trying to offer seats where there is demand for them and thus maximize their profits. An aircraft deployed on a business-heavy route like DFW-ATL or ORD-SFO on a weekday might on the weekend be sent to Las Vegas or the Caribbean, since business travelers tend not to travel on weekends, especially domestically.
But airlines can also adjust capacity and demand without changing aircraft or flight schedules. On routes and at times where there is higher demand, they can simply release fewer discounted seats, pushing cost-sensitive travelers to choose alternative flights and reserving capacity for those willing to pay a premium for it.
Yes, look at the timetable for Turin airport: http://www.aeroportoditorino.it/en/tofly/flights/timetable
It's in the Alps, and a popular airport for skiers. Some flights do only run on 1 day only, and this day usually falls during the school holidays for the departure airport of the flight.