Ok, now everything is working fine, even in Offline mode.
Here is what I did, after following some suggestions found in other threads.
First thing, I tried to repair the downloads through Origin, right clicking on the DLC packages for the repair option, in the extended Information section. It found no errors.
I then deleted this file (OnlineProfile.profile) in this folder
C:\\Users\\...\\Documents\\BioWare\\Mass Effect 3\\BIOGame\\Config\\OnlineProfile.profile
Then, whilst online, so that everthing could be authorized, I started the game and got no error message. After loading up the game, I checked and all the Firefight weapons were there, plus a mech-dog was walking about I had not seen before, and the N7 hoodie is in my casual wear (all with their corresponding messages in Shepards personal terminal). I'm on my fifth playthrough and first time I'm seeing this stuff.. heh...
After exiting the game whilst online, and trying to start the game in Offline mode, the game wouldn't start, so I had to restart my PC for Origin and the game to start up again but, everything is working now, in Offline and Online modes...