In the 1500s, as part of the Reformation, the Swedes broke away from the Catholic Church and turned towards the Lutheran Church. As part of that breaking away they published a new Bible based on the work of Martin Luther, which was written in German. They decided in the publishing of the Bible that they would keep ö and ä, and get rid of ð in favor of th, similar to Luther's Bible. You can read a little bit more here.
It's Norwegian, and a lot of Swedish people think he sounds Norwegian, too.
Yes. to begin with, it isn't considered correct to say talk Swedish. Some native speakers do, but if you don't happen to be a native speaker, this will only sound like a mistake. You should use the verb speaks. And second, talar covers the meaning of both is speaking and speaks. But if you skip the is in is speaking, you get a phrase instead of a sentence. You would say this only as a part of a longer sentence, like "have you ever heard a cook speaking Swedish" or something like that. And the Swedish counterpart would be en kock som talar svenska, literally 'a cook who speaks Swedish', because we use the participles less.
Because there is no difference between "A" and "One" in Swedish (just like there is none in German, French, Spanish, etc.).
And yes, that means that it is will be accepted everywhere where the translation with "One" makes sence. E.g. "I am one girl" is probably not going to work, because it is hard to be two girls.
No, it's because some people think talar is highbrow or formal - and while it does usually belong to a higher register, it's not synonymous with pratar and I find that most people do use both correctly without even realising they do.
In this case, you would generally tala to mean knowing a language, but you wouldn't use it as often about just having a conversation. So if your friend means the latter, they are generally correct.