In order to download the game, the one downloading it must be signed into Origin (and in online mode), while everyone else who is using the account has to be offline or in offline mode (directions to find that are listed above).
As to the "Incorrect Password" issue, in order to sign into Origin (and thus download the game), you need to know your password. If you don't, there are options to change your password which will usually be sent to your email, so I hope you know the password to your email at the least...
I had no issues letting my brother log into my account on his computer, in another city, download the game to his computer, run the game, etc.