"Could you carry my bag for me?"
Translation:Skulle du kunna bära min väska åt mig?
To answer this and the comments above I've looked up the definition of those two words. För can mean for, but it doesn't seem to be possible in this sentence, so I'd go with åt.
för for, for the sake of something or somebody = for the purpose or benefit of someone or something; to satisfy the demands of someone or something Here's a sentence from Duolingo where för is correct: https://forum.duolingo.com/comment/9172570
Jag har en present åt dig
I have a gift for you
Yeah it is exactly that. The Duo contributors actually mention that in one of the early lessons, saying that on things like Facebook chat etc. you might see "mej" and "dej" which are definitely "wrong" in written form but basically amount to how you would say the things. For English speakers it really makes no sense to learn to write them this way, because for us neither "mig" or "mej" are "phonetic" anyway, so we might as well just learn that "mig" = "may" (pronounced) etc.
But we sometimes write "nup" or "nah", or "yep" or "yeah", none of which are shorter than "no" or "yes". It's just a colloquialism but certainly wouldn't be considered correct but just like those "nup/nah/yeah/yep", it makes the whole thing a bit more casual and relaxed rather than using "Yes" on Facebook or something. If you think about how you type on Facebook, "yes" conveys a different meaning to "Yeah". Most of the time, the latter is more friendly, so I'm guessing something similar might apply with "mej och mig". For your interest, Despicable Me was released in Sweden as "Dumma Mej" (Stupid Me) but with that spelling, not "Dumma Mig". I'm guessing it's like aptly spelled, like when we say "dum" and deliberate use that incorrect spelling for irony.