IIRC, it is a real station at the airport, and you should be able to buy all the passes etc. If you take the taxi, there is a downtown transit center at the foot of Powell, sort of underground, where you can do the same and more. This station had by the way, the highest ratio of homeless/vagrants/scam artists I have seen anywhere since Times Square in 1965. I would leave the family at the hotel and send one adult to buy the passes. It isn't particularly dangerous, I think, but there were a lot of people with a tenuous grasp on reality, whether through dope or schizophrenia, and many of them were given to vocalizing.

And no, the passes do not include BART, which is an entirely separate system (the pass covers the means of transportation that are run by the Muni Railway, which includes the cable cars, trolleys, buses, streetcars, but not BART, which is run by Bay Area Rapid Transit). Unless you go somewhere outside the city that is served by BART (Berkeley, for example), you won't use BART once you're here -- within the city, Muni is more extensive, with more stops.

Hi
here is a list of all the places to buy the 3 day MUNI pass. There is a station at the Airport and at Union Square.
http://www.sfmta.com/cms/asystem/farevendpassport.php

Your hotel is about 4 blocks from the BART Powell St. Station. There is an slight incline up Powell St but it should not be troublesome. I am always awkward with the BART machines, but it's not a terrible issue. I have traveled from SFO on Bart with my 89-yr-old mother. That said, with 5 people you might find it just as cost-effective with a taxi.

Awkward is a good word.