As far I know, you need internet to download addons. I use XBOX 360, myself, but I believe they share similar concepts, in this case. Plug your console to a nearby internet link, browse and find your wanted Addon and download. Some games, like the Fallout series (another Bethesda creation) actually SELL addons. Though, I do not believe that the Elder Scrolls offer any such things. Sadly, I do believe that an internet connection is required in order to download DLCs.
As for Knights of the Nine, it works simply because it was installed before you lost internet. Any addons you downloaded before this event should work just fine.
I hope this helped. Should you require anything else, feel free to leave a message on my Talk Page.
You will, indeed, need to visit the PS Store in order to download it. Although, as I said before, I am unfamiliar with the functioning of a Playstation console. Although, I do believe that, in order to transfer an Addon from CPU to console, you require Internet.
Basically; buy console, buy game, wire said console to internet and download Addon.
That's up to you. The Last Dragonborn is merely a title, like Hero of Kvatch or Eternal Champion. Nerevarine is the only one that could be considered a name, as it refers to Indoril Nerevar.