Modern English requires "does not" to make a sentence negative. However, you might have heard someone say the equivalent of "That concerns me" (-- pause --) "not". It seems to be entertaining because the first part of the sentence implies one thing, and confuses or astounds the listener, and then the sentence is negated with "not". This type of speech is meant to be "silly" or "fun" rather than serious conversation.
It is not. "Mich" here is the accusative form of "ich" - "mich" translates to "me" in English. "Me" is a direct object, as in the sentence "He likes me". You couldn't say "Me likes he" without being grammatically incorrect in English and the same goes for German. In this sentence "das" is actually the subject acting upon the "me" - so "This/That does not affect/concern me" is the better translation.
As explained by kentel, the structure of the sentence is different, reversing the subject and object. Duolingo generally requires the student to keep the same sentence structure. Nonetheless the meaning of your sentence is the same, and if you read a published book in two different languages you will find many such "loose" translations.