Yes, the family name is Bernadotte, as they are descended from the Napoleonic Marshal. Bernadotte saw the writing on the wall for his career and wisely accepted the offer of a crown. All monarchs have a surname, the English have Mountbatten (Windsor), Monaco has Grimaldi, Germans have Hohenzollern. Some are indeed merely dynasty titles, but one may refer to them as the Bernadottes, Grimaldis etc.
True, although the 'of...' construction in English is still a genitive construction. The 'of birds' in 'a flock of birds' is genitive; one could theoretically (albeit rather obscurely) render it as 'a birds' flock'. The same is true of 'a group of men', 'the charge of the light brigade', and most of G.R.R.Martin's books' titles (or the titles of the books of G. R. R. Martin).
Not really. Konung is rarely used at all.
Actually, I would be more comfortable with your sentence if you made a swap:
Kung Carl är Sveriges konung.
"Kungen i Sverige" - sounds very odd. If you really want to use this construction you'd have to say "kungen av Sverige". But this is actually an exception. Usually this "... av ..." construction is just plain wrong. With kings it does, however, work. I'd recommend you to use simple genitive in all situations, because that would never be wrong.
No, there is a big differance between the two systems: in the UK the queen is an intrensic part of the govermental system according to the constitution and the state government would technically fail to exist with her, the Swedish King however has no constitutional power and I don't believe he is in the constitution. He definitely has no nessesary role e.g disolving parliament, letters patent, appointing clergy and aristocracy and knighthoods.
Yes and no. The Swedish king is still the head of state, and as such he has ceremonial constitutional duties that still need to be exercised for the government of the realm to function normally. The king is the one who (at the request of the speaker) declares the annual meeting of the Riksdag to be opened, the one who accepts foreign ambassadors, and chairpersons a few special goverment meetings.
Swedish doesn't have a state-sanctioned nobility, and thus the king can't knight people. The nobility families still exist, of course, and organize themselves in Riddarhuset (approx. "the house of knights"), but it's a private organization these days. The last time a person was knighted was in 1902.
Well, if you really want to use that construction, you could say "Vad heter kungen av Sverige?". However, we don't really like this construction in Sweden, and talking about kings is one of few exceptions where this actually works. In most situations, it's just plain wrong.
My recommendation is to always stick to the simple geintive, i.e. adding an "s".