I would contact Mint customer service on both of these issues. They take FOREVER to get back to you but when they eventually do, someone will fix your problem. I had a CC that wasn't compatible with mint. I sent them an email and around six months later, I had forgotten about it, and out of the blue, I got an email saying they had fixed it, and lo and behold, mint now accepted the card.
On to your specific problems, I don't know why the target card would show up duplicated. Is there anything you can change in your preferences of how it displays on target's site? I'm not sure if that would effect how mint receives the data or not, just throwing an idea out there. Since you seem to spend it only on groceries and household items, can you just make that one budget item to track called groceries/household, and then remove the target card, but designate payments from your checking account to the target card to the groceries category? Its a kludge but better than having your spending all jacked up.
On the second one, there isn't much you can do but contact mint and hope that enough enough users from your bank contact them that it gets added.
Good luck.