ups, sorry Hilda, I didn't connect back until just now (nearly 12h later) and I see Joan has already responded.
Anyway, Còrsega is a long street, busy and it'll be easy to find a taxi, even at wee hours. Yet, if you have a 7am flight to the US, and noting that your company is likely to ask you to be at the airport 3h in advance (for US-bound flights that is!), that means 4am, which means leaving the apartment at 3:30am, so I would simply arrange one for that hour to pick you up instead of hunting one in the street with your sleepy kids and carrying your luggage.
Joan has supplied the name/telephone of one, allow me to add a few more so you can have a choice: Taxi Class Rent tel.933070707 www.taxiclassrent.com ||| Barnataxi tel. 933222222 www.barnataxi.com ||| Ràdio Taxi 033 tel.933033033 www.radiotaxi033.com ||| Servi-Taxi tel.933 300 300 www.servitaxi.com ||| Taxi Amic tel.934208088 www.taxi-amic-adaptat.com ||| Fono-Taxi tel.933001100 http://www.fonotaxi.net/
Notes: (1) all taxis operating in Barcelona are bound to the same fare system, so it doesn't matter which company you choose, you're going to pay the same fares. (2) Fares are metered and depend on the day/hour being the nighttime during the weekends the most expensive, up to 30% more, which means the estimate Joan gave you is more likely to be 40-45€. Fares for 2018: http://bit.ly/BCNtaxi2018
From time to time, I personally use the first one to take me to the airport. All their cars are comfy spacious Mercedes and have different types of vehicles to accommodate more than 4pax if necessary -or 4pax with a lot of luggage!
As per how early to depart:
- Your airline will probably ask you to be there 3h in advance, but, frankly, to me, it sounds far too conservative, especially at those hours. I would probably arrive at the airport at 5, but again, check with your airline as you don't want to be left behind.
- Theoretically (theoretically!), departing on a Sat at 7am you shouldn't encounter long lines, neither at the checking nor at security -but again, that's always a challenge in all major airports I frequent, including BCN, as sometimes the less likely hour to be a problem suddenly becomes a problem, be because there are less staff available, be because you encounter several other big 777's departing at the same time, or just because the planets are not properly aligned. So there, I don't dare to advise.
Enjoy!