In a word, no.
First, start being honest with yourself. You ask "Is it feasible to do Barcelona, Cinque Terre and Paris in 14 days?" Then you go on to say that your flight departs from Madrid, a fact you omit from your question.
Next, start being honest about how many "days" you have, Do you really have 14 days? I'm guessing that you are counting one or both of your arrival/departure days. You can not really count the day you arrive in Europe since you will be quite zonked-out from being awake most of the night on your overnight flight from the US. You will also most likely be disoriented and will need that first day to get used to the strangeness of being overseas. In fact, most people are only about 75% "there" on the day AFTER they arrive, but let's ignore that for the moment.
Every day you move from one place to another, you will pretty much lose that whole day. If you're lucky, and everything works out nicely, you may have a few hours to use at the beginning or end of your day.Too many days wasted getting places.
With less than 14 days, pick fewer places. You have to include Barcelona and Madrid (both of which are worth at least a few days each). Personally, I'd actually skip Paris and Cinque Terre entirely on this trip, as they don't make any sense. Your trip is to Spain, not Italy to France. There's plenty of great stuff to see and do in Spain, and October is a great time to be there. Here's what I would do:
Arrive Barcelona, spend the next 2-3 days there. Take the fast train or fly to Madrid, then spend the next 2-3 days in Madrid. Using Madrid as your base, spend the remainder of your days doing some of the world-class cities and attractions around Madrid (Toledo, Granada, Segovia, etc.) - just don't try to do too many.
Hope that helps, have a great trip.