Hi, jakamo and welcome to FT.

It seems that you are already reasonably committed to BA (I'm assuming that is with whom you flew to SFO, and that your upcoming flight is also with them)? There is an entire forum dedicated to discussion of BA here with a very useful primer here. In short, to answer your questions:

1) Collect miles to spend on free flights and upgrades, with enough flying get promoted to silver or gold card status which has additional benefits (lounges, priority check in etc).
2) No, but you can earn the credit card miles if you're paying for them
3) Yes - get the more expensive Amex one - this is in the primer linked above
4) Depends on how much flying she's likely to be doing!

The decision of which airline's frequent flyer scheme to follow is a complicated one determined by (a) your flying pattern and (b) what you want in return. If you're normally flying in paid business or first then BA is quite good. If you're normally flying in cheap economy tickets it's not so good, and I'd advise you to consider United who make it much easier to upgrade than BA. If what you want is free flights, then British Midland are one of the strongest (which would involve flying LHR-SFO on United).

Hope that points you in the right direction...