"I don't care about myself" doesn't mean the same as "I don't take care of myself". In other exercises the same verb has been used in the sense "look after your body/heart".
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.
The suggested answer when mousing over is "take good care of". Adding good got it marked wrong. Is good implied in dbam or not? Curious.