SimCity will be getting an offline mode very soon, Maxis revealed through their official blog.
They're still testing it, and the DevTesters will get to try it out first (they're basically hardcore volunteers). Maxis say they want to make it fully functional before releasing it publically. And yes, all of your DLC will also be accessible offline.
This also means that you can mod the game without affecting the online game. However, there are still some rules to follow. They're mostly simple things like "Don't harm other players"and "Don't inflict copyright infringement", but you can read the full list of rules here.
Maxis promise to release information outlining the offline mode and some tutorials about how to mod the game in the near future.