Iceland passes the "Wikileaks law". What will it mean for journalists? (opens in new tab)

Yes. Birgitta Jónsdóttir is properly referred to as Birgitta Jónsdóttir or just Birgitta, or by her first and middle name if disambiguation is required. Jónsdóttir alone is meaningless as far as name goes. Icelandic phone books are ordered by first name.

edit: although to be fair, in the context of an article written in English aimed at a general audience this is not an easy situation. Properly using just the first name might throw off people who don't know about the Icelandic conventions (that would be most of them) who might then consider this use as strangely personal. On the other hand, repeatedly using the full name might look weird too. Personally I'm not convinced these are worth the trade-off of being culturally incorrect, but I'm not the one making the calls.

The reason you'd call someone Mr. Smith in the English speaking world is because surnames are culturally significant. Because they group entire male lineages together.

That's not the case with patronyms, so this usage has never developed. In Iceland you'd never use just the patronym. It's not a proper name, just an indication of parental lineage.

Many have some sort of same-sex unions, what Iceland did recently is eliminate all occurrences of "man and a woman" and replace them with "two persons" in their laws.

Most countries that have same sex unions have a different set of laws for them, usually for religious and political reasons. The Church of Iceland opposed the recent change, but it lost that battle.

Regarding Jóhanna, it's sad that the first openly gay head of state is also so openly incompetent :)

I thought part of the problem there is that, while they declared themselves independent, nobody would recognize their independence, thus they were still effectively under the laws of the area they were in.

If Iceland, an already-recognized nation, decides to do this, trying to get things shut down without their consent could / would be seen as an attack, and it becomes a far larger issue than simply enforcing laws within your jurisdiction.