ebony,
to answer some of your questions, first the easy ones. from the blackbushe car park to an aircraft on the apron, no more than 2 mins walk. no, you and the kids do not need passports.
now for the negative stuff:
as mentioned above, most pilots of light aircraft will not be able to accept payment for a flight, or a benefit in kind, as you suggested. without going into all the legal stuff, its almost always a no no.
secondly, even if you got yourself a PPL, you would run into weather problems quite often, so would end up driving anyway. the hours you want to fly would for most of the year be after dark, so would need a night rating, and then you run into problems of aerodrome operating hours and runway lighting (or lack of it).
i havnt got the book to hand but from memory cromer (northrepps int airport) is a short grass strip of about 500M, and not suitable for many 4 seat aircraft.
i can understand your desire to get out of the car journey, but doing it on a regular basis in a light aircraft is a non starter. thats not to say that somebody on here may not be willing to give you a lift once in a while.
good luck
CJH