Option 1: Play on the Stardock servers
Player loads up Elemental, chooses Multiplayer, selects “Host” (or Join). If they select Host, it will find the nearest (geographical) Elemental server run by Stardock and put them in the lobby. From their point of view, it’s almost exactly like any other game. But in reality, the player is now part of a client/server setup where Stardock provides the servers.
If the user chooses “Join” they instead get a list of “Worlds”. These are servers that have at least 1 human player already in them. The display will include the map name (random or a specific name), host, and ping time.
Option 2: Play on Custom servers
By default, it’ll be just like the Stardock servers. But these servers, run on third-party machines, can be highly customized. Players can mod their servers with custom content, custom rules, and lots of other things.

The above information, that we can have our own instances of servers, might indicate that we have the abillity to do single player offline. I'd like to know if we can do option #2 offline, on an internal lan, because I spend a good portion of my time in rural towns that just don't have access to broadband.

This isn't Ubisoft. Internet will only be required to play multiplayer over Stardock's servers and to update the game. If you want to play single player or LAN you won't need to be connected to the internet.

EDIT: You CAN update your game if you don't have the internet on the same PC as you do the game, but can use the internet somewhere else, it's just a little more complicated then just opening Impulse and clicking update. I've never actually done it, so I'm not too sure on the details, but it is possible.

Internet is not required for single player.  However internet is required to play the beta currently (it makes updates and data fixing easier).  So best to wait for release.

Updates without internet are always a pain, regardless of game or system. And stardock has the best DRM around(the kind that lets you do whatever, whenever), so no worries about that.

Simplest way to get updates this way is to use a computer that has internet, and install Elemental. Do the updates. Use Impulse's "archive" option to make a copy of the game, stick it on a USB drive or something.

Go to your offline computer, restore the archive. Ready to go!