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2) Consulting recruitment (1)
If I interview for consulting (McK, Bain, Booz, BCG) during 1st year (i.e. for summer internship) and get dinged, can I still apply for FT recruitment? Or it's 1 shot only?
You definitely can. In fact, historically the number of full time offers in consulting has been higher than the number of summer offers - that it to say that many of those who did not receive summer offer received full time offer (sometimes at the expense of those who received a summer offer but did not fare well during their internship.
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3) Consulting recruitment (2)
If I study in the US and interview for foreign office (Europe) do I have to fly to (or have video/phone interviews with) this office? Or maybe recruitment is done by US office and they will send their decision to the foreign office?
Short answer is: It depends. Usually you still have your interviews with global affiliates in the U.S. (some partners may fly over to, say, NY and interview applicants from many schools during one day)
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5) Business cards
I've heard recently that students order business cards. Why? Do you hand them out during company presentations etc.? If so, what for? Doesn't it look cocky?
Yes, they do. Same reason that you have your "usual" b-cards - to be able to share your contact info. Yes, if it feels appropriate. No, in most cases it doesn't look cocky. In fact, you are expected to offer your business card sometimes.