First of all I'd definately purchase an open jaw ticket flying into one city and returning home from another to avoid backtracking. I honestly think you're going to need to cut back one location. The time in each city isn't factoring in travel time between or time to recover from jetlag. For example, 3 days in Barcelona will really only be 2 days as the first will be jetlagged (assuming you're arriving on a transatlantic flight). You could easily spend 4-5 nights in Barcelona alone. 4 days in Paris is probably my personal bare minimum but you'll loose at least 1/2 day arriving from Barcelona. Yes, this can be done logistically but it's covering alot of geography and countries in just over 2 weeks.

You're in your early 20's and will have plenty of time to return!

Beach resort time!! Take a bus to Lloret De Mar from Barcelona for a few days, then back to Barcelona.. We stayed in neighboring town, ( tossa de mar) but its much smaller and quieter, but on the bus we took all the younger folks got off a Lloret de Mar.. looked like fun if I was in my 20s.

Paris, then Amsterdam..

With 16 days you have to sort of subtract your travel days , so really it end up being about 13 days.. three major cities is enough,, plus there are so many fun side trips.

Make sure your two days in Amsterdam are two full days, not arriving one day and flying home the next. Two days is the absolute minimum for Amsterdam imho.
You would be better calculating in terms of nights really.
Don't forget also that if you are from the west coast of the US your flight arrives in Europe the following day.

Thanks, folks!
I did factor in all the travel times to-from places, so the 16 days are full days we will have in Europe.

After having done some research, I agree it's better to take it easy and limit ourselves to less destinations. Hopefully my friend will agree! I will look into Lloret de Mar, I've heard lots of good things about it. Paris would definitely be the next logical stop then? We've already ruled out the French Riviera, so I think Paris then Amsterdam should be good. (London would be too much of a hassle, eh?)