Hi,

This past summer we paid 30 Euros with a pre-arranged cab from Barcelona Taxi from airport to hotel in city center . Later we used a regular cab from/to the airport an it was around 25 euros (maybe a little less).
The port is very close to city center. I went years ago but can't remember what I paid but it couldn't be more than 15-20 Euros as it is closer than the airport.

I am not 100% sure , but I think I researched this once and if you fly other than disembarkation day and do not use their hotels you are cahrged the deviation fee.

Are you staying 2 nights or more? The Majestic shows 274 Euros per night, including breakfast and Tourist Bus pass or entrance to 6 museums if you book 2 or more nights.

Of course, The Majestic is not cheap , there are better priced hotels but might not be as nice.