Yes, but the Polish sentence covers both. Judging by the context (or the lack of it) „Nie dbam o siebie” is rather the first. If one is describing someone else it's very probable that one means the second one. Often this first meaning is expressed as „nie obchodzi mnie to, co”.
„Nie dbam o to co się z tobą stanie” - I don't care what will happen with you.
In ref. to 1.: So, it is the same concept as with the possessive pronoun “swoj”, that it automatically refers to the subject? Understood, this should have been obvious to me.
In ref. to 2.: OK, I thought that it would just be as with the English “fine”; with the adjective it sounded clumsy to me, unnatural even. But I was seemingly wrong with this idea. Thanks for correcting me, much appreciated, as always! :-)
You can't think of prepositions as something that can be translated as 1:1 equivalent. The main meaning of "o" may be "about" (on the topic of), but as most prepositions in any language, it has several other usages, this is one of them. You should rather learn collocations than just prepositions.